Kali's Blog on rekting in the Arena.

Kali has many different ways to be built, and hence, multiple functions she can function as.
Most people see Kali as an endgame hypercarry in conquest only, but she has her perks in different gamemodes too.
In Arena, she works by catching unaway opponents, or just simply someone who's slightly further off from the middle of the warzone, and destroys them like the fodder they are.
Thanks to her passive, her regeneration is pretty impressive when she gets a kill or assist, and oh boy does she get them kills (unless a teammate is doing some KS stuff... bad memories right there, got me killed a bunch of times)
Her being aa focused, means that her early game is usually weaker than other assassins in the same position (take Thanatos's halfhitting 1 for example), but her midgame picks up, and by lategame, she's the brutal anus destroyer we all know and love.

Many of my builds focus on different goals, and can be used more effectively than the other ones against certain opponent combinations.
The build that include 2 crit items are usually good against opponents with 1 tank 4 squishy; 3 crit items for a full squishy team, and more shred instead of crit against more tanky opponents. It's kind of self explanatory, but I tought I'll point it out anyway, since some older guides of mine got some criticism due to similar reasons.

CORE
Ninja Tabi, Hastened Katana, and Quin's Sais. These 3 items are generally there on literally every Kali players, here's why.
Ninja Tabi: You'll need boots. Everyone needs boots. Movement speed is important, you can't kill targets, if you can't get to them.
Hastened Katana: again: movement speed. If you get combat movement speed reduction, sticking to an opponent becomes that much harder. But since you get that debuff removed for 1-1 sec with literally every basic attack you land, you don't even have to worry about it.
Quin's Sais: shred. If there's 1 item in this game that can shred opponents with high/somewhat high protections easily, it's this one. The higher their max hp, the more damage they take, regardless of their protections, due to it's passive. Also by the time you get her in your build, you'll already have a nice base attack speed, even if you'll be a little lacking in power.

CRIT:
there are generally 3 crit items that are often picked for her, but let's examine all 5 possiblilities. (Also these can be great assets for getting achievements for her.)

Deathbringer: Everyone loves it, gives high power spike, 20% crit chance, and mst of all, +40% crit dmg. Pretty handy if you crit often enough, if you build 2 or more crit items, I'd definietly include it.

Wind Demon: speed, attack speed, crit chance. what else do you want from an item? Alongside Executioner, this is also a pretty common pick, it works especially well with Hastened katana's passive. Ever seen a Blue ball of death charging your way, running you over, and leaving you dead? That was probably Kali using this.

Rage: extra high crit chance among crit items. As a first crit item, it's definietly recommended, if you only want to build 2, since you'll have about as much crit chance with 2 crit items, as you'd have with 3 other ones.

Malice: since you're mostly focusing on destroying your opponents by the simple amount of attacks you have, and not with a huge powerspike, this might be less effective, yet it's still funny to see an opponent get away with a movement skill on low hp, and die to the effect of this item. Definietly worths it if you're ahead by a lot, and want some big lols, but otherwise I wouldn't necessarily recommend it.

Poisoned Star: a reasonable pick in terms of confirming kills, or possibly giving support to your team, but gives you yourself less of an advantage than Wind Demon would. While this way you can decrease opponents damage output considerably, and still smack them with almost as high stats as you could with wind demon, but a different passive. In terms of confirming kills, it might be better, but it's somewhat less likely that it'll be you who gets it.

POWER:
you won't really need much of it, you do shred your opponents a lot after all. Though it still might be useful for crit builds, if you want to make some hybrid setups.
other than the power you gain from above mentioned items (because quite some of those hold a reasonable amount of power to come with those stats of theirs)

Transcendence: the highest power item in the game (currently). It can proove to be a fix for your early mana hunger, but in terms of late game, you wouldn't have that issue anyway, and while powerspikes help everyone, attack speed, lifesteal, and crit chance helps you more.

Shifter's Shield: while it can most certainly make you less squishy (against physicala targets) and give a good powerspike, that's where it's effect ends. And if you use your ultimate, you can't die anyway, so that little bonus to your deffense won't do much. Especially that mages and guardians can still do just as much damage to you.

Bloodforge: While it is a huge powerspike, and that's good, I recommend it for a different reason. Bloodforge has 15% lifesteal, that can be extremely helpful while taking out opponents, and you will definietly like seeing how you get your hp that they take, back to where it belongs. There are definietly better lifesteal items out there, or at least ones that are just as good as this one, but if you want power, this is practicly the best choice I can give you. Not many items can ige more than the 75 power from bloodforge anyway.

LIFESTEAL:
you like killing people? okay. You dislike dieing to them if they hapen to be a tad bit too hard to chew? get some more hp, and you won't have this issue.

Devourer's Gauntlet: while it's a pretty nice lifesteal item (stacked, it's probably the highes lifesteal one in the game, unless Asi's passive is up). Overall not the worst pickup, but stacking it takes long, and it's not exactly Kali's forte. especially early game.

Bloodforge: less lifesteal than Devourer's, but gives more strength, and doesn't need stacking. (even if it's more expensive). If you go lifesteal (and you probably should, unless you're going for achievement) this is one of the best pickups you can have.

Soul Eater: the new version is less trash than the one rewamp older one, but now it's more of an item that supports ability based characters, not for aa based ones, like Kali

Asi: probably the best pickup for Kali, since it gives penetration, attack speed, and lifesteal in 1 item, making her much more lethal than whatever she might have been before. Also it's passive can be a mayor pain in the opponent's butt, especially if you can follow it up with your ultimate right before you'd die. Your oppoents will not dare mess with you if you do that once or twice in a teamfight.

SHRED
your main source of damage, the more you do, the less your opponents will dare come close.

Quin's Sais: As explained above, it rekts. Especially so on a goddess such as Kali.

Executioner: getting 3 hits in with such attack speed takes but 1.5 seconds (1.2 at max atkspeed cap)and giving you more penetration than Titan's Bane is definietly worth it. Not to mention the power and attack speed bonuses. it's a really good item to have.

STUFF TO SUPPORT YOUR TEAM
As mentioned above at Poison Star, some items doesn't make you better because it increases what you can do, but because it decreases what your opponent can. Poisoned Star is one such example.

Poisoned Star: decreases opponent's damage output, and movespeed. helps in confirming kills. Overall not bad, but to get the effect off relyably, you have to have at least 1 more crit items, so it somewhat restricts your build.

Frostbound Hammer: if the opponents can't run away, you're all welcome to finish them off. Sometimes it comes in pretty handy, also that health bonus makes you somewhat less of a squishy.

As for every god and goddes, there are 2 achievements for Kali as well.

The easy one:
Kill your marked targets 5 times, in 1 game. Hint, if you go on a mass mudering spree, always go for the closest opponent after you killed your first target, and you'll probably get multiple stacks in the achievement's progress. Because mark gos to the closest opponent anyway. Or only living one if that's the case. If it's a Deicide, it goes to the opponent who spawns first, or closest to you, when they spawn.

The less easy one:
Get 2 kills in your ultimate while below 10% hp, in 1 game.
Only harder, because you'll either have to be really close to your opponent in terms of damage output when you do this (and pop your ult at the right time) or simply popping your ult when you're almost dead, and hit them with everything you have, so they die before your ult runs out.
As a hint, for this achievement, I'd recommend using Thorns. In a teamfight. Pretty handy. Might get a kill even if you're targeting someone else simply by getting in their way. And if you're already on 1 hp, you don't take damage anyway, while it reflect is before damage calculation, meaning your opponent still does take it.
In Arena, getting this achievement can only be hard, if your team steals it, or if your opponent runs away once you popped your ultimate. That's why you keep your 2 off cd there, to throw it after them. Also having your 1 off cd can help too.