Swiftness Boots
I would almost always take swifties. There's certain matchups where it's absolutely necessary, but in general is is REALLY nice for letting Tahm Kench stick on enemies, which he loves. It's also great for escaping, as even with W up, you often need to make distance to safely use it to escape. Aside from all this, the slow resist helps you drag enemies further with your ultimate. Ninja Tabi is a bait I think in almost every situation. I might take it if the legit have 5 auto attackers, but even then it isn't going to block any BotRK damage, or any sort of on-hit, which is the sort of thing you have to watch out for with auto attackers. Merc's might be fine against really heavy CC comps, but again, I much prefer Swifties in nearly every situation. It's cheap, and it lets you stick on enemies, which TK loves. Oh, and Sorc Shoes can be nice for carrying if you're really ahead, mostly against squishy comps that also won't catch you out since you'll have less movespeed.
Oblivion Orb
Oblivion Orb is my preferred anti-heal choice. I think Bramble might be fine vs a healing heavy AD champion that you only want to survive against the entire lane. I think Renekton and mayyyybe Sett are the only matchups that really come to mind. I think the little bit of AP you get from Oblivion Orb is really good with the health you build (TK passive has AP scaling that is higher based on how much health you have). Not only that, but Oblivion Orb can be procced off any damage that you do, since your passive also does magic damage, whereas Bramble is only activated if you're autoed. This means even against the ideal champions, Sett and Renekton, they can just heal off minions and ignore you. I don't recommend ever building this into Morellonomicon unless it's your last item. Most of the value in Morellonomicon is from the anti-heal, which you already get from orb. The only time I'd think about building this early is if I want more AP but don't wanna build a different AP item, but I think even then it feels quite wasteful.
Doran's Ring
This is a bit counter-intuitive, but I would actually pick Doran's Ring in extremely difficult matchups, such as Vayne. The idea is that you sit as far back as you can, only using max range Q on minions (or the enemy to damage them and heal yourself if they walk up). With Ring, you won't run out of mana during the early levels, as you usually would with Shield. Doran's Shield is a great default though, as it synergizes very well with Second Wind and Tahm Kench's E ability. I usually take Doran's, unless I really need the extra mana early and am planning on playing very safe. I think there's an argument to be made for Ruby Crystal start too in matchups where you can't kill the enemy but aren't in danger of dying either.
Elixir of Iron
This is my secret sauce. I've been vouching for considering Elixir of Iron after 2nd or 3rd item for a really long time now. For just 500G, it provides 300 HP (which alone is 800G when bought [though Elixir is a rental]), 25% tenacity, a 15% movespeed trail for allies, and most importantly 15% increased size, all for 3 minutes. IS IT WORTH IT? I think in situations where the Q range you get from size is valuable, without any doubt, yes. If you consider this to be like borrowing a full item (which I do, the Q range is insane), then buying 5 of these throughout a game would get you 15 minutes of elixir for 2500G. That's a really cheap item, and 15 minutes is a long time. Remember that TK wants to end games early. If you can spike and stay ahead of enemies, sometimes that is what you need to win. I will add though, that you should take note of the enemy. If they have good engage that won't really care about your Qs then the extra range might not matter as much. Also, the crazy range it gives can allow you to stick to your carry better, so that you can eat an ulting Twitch while still poking enemies. This feels really nice after a couple items before big objectives though, I'll tell you that much.
Heartsteel
Heartsteel is usually going to be the item you rush. This item feels great when ahead, it feels great against squishies (because Tahm Kench's health scaling passive, and Heartsteel's passive both do flat damage), and it feels great in matchups where you need the extra tongue speed or range (the range also can allow you to stick on your carries easier, ready to eat them while also contributing your Q to the fight). It can feel not as great when enemies have a lot of health and a lot of %HP damage, but you should probably still at least consider it due to the health and size it provides.
Bami's Cinder
Bami's isn't what it used to be for TK. You used to always have to rush this before anything else, but the game has changed a lot. I think the Bami's items, Hollow Radiance and Sunfire Aegis, can be great second item choices vs champions that would be oppressive in sidelanes, such as Sion or Garen, or even first item against high damage wave shovers such as Mordekaiser or Gwen.
Warmog's Armor
Warmog's feels great as a second or third item when ahead, mostly against comps that don't have a lot of CC. If they have a lot of CC, it is easy to shut you down before you can regenerate your HP, and then the game will be a lot more difficult. I recommend considering Warmog's if they have squishy pokey champions with no CC, but in general it makes a nice core item.
Riftmaker
Riftmaker is great when ahead, especially if you built Heartsteel first. A little bit of AP with a lot of health is something TK loves, and Heartsteel gives you more range to make use of this extra damage, since it'll be harder to fight in the middle of teamfights when you don't have many resistances. Consider elixir of iron right away if you have the gold after Riftmaker, as it provides a bunch of range.
Unending Despair
Great core item in general. I think it works nicely as a 2nd or 3rd item, especially in comps where it is easy to get caught, such as if they have a lot of CC. Imagine Pyke pulls you in: Do you want Warmog's in that situation, or Unending Despair? Unending Despair will allow you to survive in the middle of teamfights longer, so I think the idea is that you build it when you can't avoid that danger.
Rylai's Crystal Scepter
Bear with me. It's true that Rylai's provides NOTHING for Tahm's kit. His Q already slows for 2 seconds (and the slows don't stack), his W already knocks up for 1 second, which is the same duration as Rylai's, and his ult can be already combo'd into Q with correct timing (see my tips page). So why would I pick this item? Well, I'm not sure if it is meta, as I'm the only one who has really tested this build (though, I've had good winrates with it in diamond). But if you pair it with Unending Despair, it'll slow everyone on the Unending's 4 second drain procs, making it incredibly easy to stick on enemies. I really like this when there's a Fizz in the game, because it is the one time I can ever catch him. Aside from all this, Rylai's is pretty cheap and gives stats that TK enjoys. When I take this build, I go Unending>Rylai's, and then I strongly consider a Jak'Sho after.
Iceborn Gauntlet
I'm lumping both Iceborn Gauntlet and Dead Man's Plate in this paragraph, as they both accomplish similar things. I think these are nice first or second items into difficult AD matchups, and especially ones that are otherwise difficult to escape or chase. In terms of their passives, Dead Man's helps you escape easier if you get ganked than Iceborn does, but imagine on the other hand your jungle ganks for you, and then Iceborn slow will allow you both to deal damage to the enemy. These are both nice for survivability and utility. The slow resist from Dead Man's does work with Swifties, but it stacks multiplicatively. This means that, though they both give 25% slow resist, you'll end up getting 43.75%, and not 50%. (If you have 25% slow resist and build another 25%, it'll give you 25% slow resist of the remaining 75%.) This is still nice I think, as TK loves his slow resist, for his ult and just for moving around a lot. Could be nice late game item pickups too if you need extra movespeed for killing annoying kitey champions.
Redemption
Redemption is very nice against tanks. It's super cheap, and does 10% max HP true damage, which is unavoidable as long as you time it right with your ult. I don't recommend it against targets that can heal sustain this damage well, but I think it might be a nice answer to a champion such as K'Sante, as he loves to spam magic resist and health. Keep in mind though that TK loves popping squishy champions, and by rushing Redemption you're sacrificing Heartsteel's ability to do that, along with its range. I do think Redemption might also be a nice answer to Kindred, both on enemy and ally team, as you can time it with the end of her ult to give all allies healing while damaging all enemies.
Dark Seal
Dark Seal is a very nice pickup, perhaps every game. TK's passive scales not only with health, but also AP based on how much health you have, so in other words a little bit of AP with a lot of HP is really nice. In some instances, I might consider upgrading this to Mejai's, as it allows you to snowball really hard, and stay ahead, which is important as TK, as he tends to fall off hard after enemies reach 2 or 3 items.
Knight's Vow
Knight's Vow is really great with melee carries, such as a fed Samira, Nilah, Viego, Graves, and other similar champions. It can also be really nice if the enemy team has assassins that threaten your carry. This is a cheap item too, which is nice, but I would only build it if your team is doing great, and they're your win condition. If you need to be the carry, I would not build this.
Spirit Visage
Spirit Visage is great against AP enemies when you have some good enchanters on your team. If I see Ivern or Soraka on my team, I ask myself if the enemy team calls for Spirit Visage. That way I'm not only getting extra healing on my abilities, but also their abilities. Lots of value.
Frozen Heart
Be careful with this one. It provides a lot of mana, which is useless on TK, and it provides 0 HP. I think there's an argument for using this against Kalista or Vayne, but I think it is very difficult to make this item feel efficient. Maybe as a late game item against champions that would suffer against it, when you have a lot of health, or maybe even early if you're really high elo and your mission is ONLY to survive and be useful for your team (iron-diamond games are a lot less stable then top 1% games, and you can't really trust your allies to make use of this).
Titanic Hydra
Nice item if you need AD on your team, and has great waveclear. In general it feels pretty nice after Heartsteel. I haven't done a whole lot of testing with it, but I certainly think it is a viable option, especially if you plan on split pushing or responding to a split pusher a fair bit. Did K'Sante build a Kaenic Rookern? Maybe he'll regret that purchase if you start doing good AD damage and pushing him in. Again though, I haven't tested it a whole lot.
Abyssal Mask
Abby Mask is very nice when there's AP on both your team and the enemy team, especially if the enemy team has beefy close range champions. I think it pairs well with other items that allow you to be closer to teamfights, such as Unending Despair and maybe even Jak'Sho.
Jak'Sho, The Protean
Jak'Sho is nice when paired with other resistances. I don't think it fits well in a standard Heartsteel build, but if you go something such as Unending Despair and Bami's item, then Jak'Sho ends up getting a lot more value. Might be a good idea in situations where you can't avoid being in the middle of a teamfight very easily.
Cosmic Drive
I think this item might be underrated, and could use a lot more testing. I haven't experimented with it enough, but I think AP, HP, and movespeed are all things TK LOVES, and ability haste isn't terrible either. It's also hugely gold efficient. Yes, part of that efficiency is in the ability haste, but overall it is 128% gold efficient, and increases to 136% gold efficient when the passive is activated. (Gold efficiency can be deceptive, because some items have passives that are really valuable but simply its worth can't be determined with gold, so take this with a grain of salt. The point though, is that TK is a stat-check champion, and loves most if not all of these stats.) I do think this is an item that takes a specific playstyle, so it is hard to judge it off winrate alone, given its low popularity.
Randuin's Omen
Randuin's Omen is a nice option against crit-heavy enemies, and the slow is nice vs melees too. If they have Yone, Yasuo, and Samira, then I'm certainly thinking about building Randuin's at some point.
Winter's Approach
Fimbulwinter (which turns into Winter's Approach) is a cheap item that gives really nice stats and shields, but I'm actually putting this here to speak of its issues. The tear takes a really long time for TK to stack, and this item hinders your early spikes, which are really important on TK. I REALLY like the idea of Rod of Ages into Fimbulwinter, but unfortunately it is just too difficult to consistently pull off with the spike delays I think.
Rod of Ages
ROA, for the same reason as Fimbulwinter, is a bit of a rough one. I think in situations where you don't need to worry about having a good first item spike this might be fine, but I think that situation is so rare. Your first and second item spike is really important, and this item takes a long time to come online despite how cheap it actually is. I think ROA is actually fine if you build Heartsteel second, but I'm not convinced it's optimal on TK, especially in high elo.
Zhonya's Hourglass
I believe Zhonya's might be good against heavy AD comps when you can eat good combo champions. For example, Fiddlesticks or Swain might already build Zhonya's, but Samira doesn't. If you eat Samira while she's ulting, and then Zhonya's, that might work very well! Furthermore, it works well with waiting for your Q and E to be back up, so you can snowball more health. Since Hourglass duration is 2.5s, and your ult duration is 3s, you can also use it to eat an enemy and then combo into your Q without taking any damage.
Other Items
Really there's so many options, and I'm all for creativity. Kaenic Rookern is probably fine vs really heavy AP comps. Hextech Protobelt is silly but provides a little extra mobility with good stats, and I find that idea interesting. Bloodletter's Curse is probably nice when you are against high magic resist champs with a lot of AP on your team, but I haven't tried this yet. Wit's End is a funny MR split-push item, but I've only found it to be a good idea once (in a game with heavy AP on the enemy team, where I also didn't wanna join team-fights). Hullbreaker is a funny split-push item, especially when paired with Heartsteel and Titanic. Horizon Focus might be funny specifically vs Fiddlesticks. Usually I find Heartsteel, Dark Seal, and Swifties, into either Riftmaker, Warmog's,Unending Despair, or Bami's item the most useful, but there are SO many options in the game, and by being creative you can sometimes find ideas that work very well in situations that would otherwise be tough. Don't be afraid to test after you get your baseline!
