Start with [Boots of Speed] and 3*[Health Potion].
Get 1 [Doran's Blade] at your first back if possible, then [Wriggle's Lantern].
Don't forget always to get 2 or 3 [Health Potion] when you back. Jayce needs some sustain early on, especially before you get the Lantern.
Get either [Ninja Tabi] or [Mercury's Threads] depending on what you need.
Get the [Trinity Force]. You can either get first the [Sheen] or the [Phage], depending if you feel more like you'll need more flat Damages or a CC and some extra Life.
Depending on the situation again, get first the [Warmogg's Armor] or the [Wit's End] depending if you feel aggressiv and needing some MR, or lacking of HP and sustain, and then do the other one.
The next item and the item that will get over the [Wriggle's Lantern] really depends on what you need and feel. I suggest [Randuin's Omen] and [Youmuu's Ghostblade]. Omen gives you the extra life and armor you need, some CDR and the activation on it is always useful in teamfight. Youmuu is for extra burst damage, CDR and Armor Pen if you need it late game.
You can also do others last items, tbh I didn't get to lategame many times. You can also try [Atma's Impaler],[Force of Nature],[Guardian Angel],[Zeke's Herald],[Aegis of the Legion] and [Shurelya's Reverie].
Skill Sequence and General Gameplay
The whole thing with Jayce top and the E-maxing build is that his E's damages are MAGICAL and on %max health !
This spell always costs 40 mana, and has a kinda short CD for the damage it deals. He also has a longer range than your Hammer form's auto-attack.
The thing I usually do is harassing in Cannon form wirth auto-attacks early on, and as soon as I get the 3rd/4th level, I use the Cannon form's E to speed me up, switch to Hammer form, Q + E and get back. That's a very effective harass. Always be very aggressiv. Cannon form while your Q and E's CDs are down on the Hammer form, and as soon as you can just jump on the opponent and harass him.
Lately, use your Q + E on Cannon form to harass before the teamfights and boost your team. Use your W as soon as you can on any form. As a bruiser, you should be in the fight in Hammer form and trying to deny the carries.