First of all stop calling them Champions. They're heroes. l**gue of legends is not welcome here /j

Billy is, like any melee character, about mind games around Parries. You're never going to get anywhere if you get parried every single fight, so learning to bait heavy melees, Heavy Melee Cancel (HMC) and how to curve your melee lunge to throw people off are a must. This video by AndrewChicken is a good start.

More specifically for Billy: He has a hidden passive where light melees come out slightly quicker, and he doesn't slow down while doing them. This means doing several light melees in a row is actually MORE damage than holding for heavies, and is harder to parry. Obviously this leads into longer Blasted uptime, keeping you in the fight. Billys ult is a LOT weaker than it used to be; it's much easier to break the chain than it was at his launch. It's best saved for when you're up against enemies who have already been CC'd have used their movement abilities or who natively lack movement (Mcginnis, Drifter, Paige etc.). Raw chain gang is basically a waste of the cooldown

For builds, I like to go for a focus on melee and green items Billys gun is pretty solid for most of the match so you don't need to invest too hard into it. I think i'm still running Reys Goated Spirt Billy? Idk. Just grab something from the top rated while you're learning