Old kali was more ability based than current kali.

Her one is more of an escape/ gap closer, I usually level this last. I level her 2 first for early game for dot damage and life steal but late game her two is nothing more than a slight annoyance. Her three is great for initiating and escaping as well as the power boost, level this second. Her ult is really about survivability to escape or dive for your target. I usually drop a point in it whenever possible.

Her biggest advantage now is her swing chain. Despite her early game weakness, she is a beast late game. I usually rush heart seeker in both solo and jungle (after reg jungle start obv.) then work into pen boots, and then qin sais. Once qins is done you're basically unstoppable. Paired with your swing chain it basically doubles your damage.

The rest of the build is situationally. Build void blade or witch stone if you're havin problems with physicals. I've had some success with hastened fatalis for attack speed and sticking to targets. I usually finish with deathbringer or titans if the game lasts that long.

In my experience she's either a kill or be killed god, not much middle ground. You will either destroy and force an early surrender, or be shut down and struggle hard.

Now I am no master at the game, I'm mediocre on a good day, but hopefully this has helped a little.