In summary, Jayce is best played as a burst AD-caster. He's got a talisman in his pocket and his single disengage makes him a good assassin in teamfights that no one really expects.

He struggles against hard engage and is easy to lock down-- so if the player isn't good, and the team is relying on him to carry, it's not hard to shut him down if he can't kite well.

In laning phase, he's an absolute bully and is very difficult to deal with if he's mechanically good. If he hits all of his abilities, it's usually enough to 100-0 any champion (even if they build tanky) early game.

The true trick to playing Jayce is usually how you fire your Q. No one really talks about this, but if you Q-E in cannon form, and your gate isn't on top of you when you've Q-E'd, you're doing it wrong.
You're supposed to Q, then place your E DIRECTLY in front of you, or to the side of you directly after firing your Q. This gives you an instant speed bonus which is extra good for disengaging/kiting and catching people.
It's exceptionally hard to master but once you crack it down it makes your Q seriously hard to dodge.
After landing Q, just try to get an auto off or full cannon W, then proceed to beat them up in hammer form. If they are squishy they're probably dead by now, if they aren't, they'll probably be exceptionally low.

Play the role of a ADC, just poke and poke, and practice good placement. In teamfights, it's best to stay in the backline and protect your backline with your single target disengage, and when their backline pushes up, just melt their squishies. If you build this build, you'll literally just be 1 hitting everyone. The only thing I can name that counters this build is thornmail-- and all it takes is just not to focus the enemy player with thornmail.

This is literally the 'Korean build', playstyle is inspired by Narakyle. Probably one of the best versatile builds for actual good Jayce players.