For the speediest Kali, I recommend building for Talaria Boots instead of Hastened Katana. It gives you more mana regen to stay in lanes longer, so that's a good bonus.

You can choose whichever relics go first. I recommend beads for the cc, but if you know who you're going up against, and you know how to play smart, get blink first. It can get you out of a chase or a kill that almost got away.

This is a build I finish with very often. The order these items are in is not definitive, so do what suits your needs down the line. I do not recommend building Bloodforge first.

1st item - Hastened Katana for early chasedown

2nd item - Hydra's Lament to hydrate mana/burst potential

3rd item - Stone Cutting Sword to stack protections against physical gods

4th item - Executioner for that glorious attack speed and penetration

5th item - sell Attacker's Blessing for Qin Sais

6th item - Bloodforge to keep you alive when you start eating everything

(5th and 6th item are interchangeable)

Asi seems to be a product of diminishing returns for Kali, in my opinion. She already has lifesteal in her kit, and Bloodforge covers most of the lost health while giving her more physical power to play with. It's still an option nonetheless. Toxic Blade is great against an aggressive healer (Aphrodite) or healer-based teams. Golden Blade is an underrated item for Kali. It opens up multiple kill possibilities, also the attack speed added is very nice. Ichavial's nice when dealing with multiple auto-attack gods.

I saw that someone had a build list that included these items for bursting potential. I wanted to add the concept to this build sheet, and leave it open for people to consider. I've used Hydras with Crusher and it's a sweet deal so far.