you can think of jayce jungle as kayn, but for weirdos who play jayce jungle (me). you can adapt to be whatever your team needs. yall need to snowball early? assasin. you need a frontline? HP bruiser. you need consistant dps? ad bruiser works wonders. jayce has such a flexable path that you can be whatever your team needs.

there are 2 choices every jungle main must make, to gank or not to gank. Jayce's clear is slow at first but it picks up really fast, if you need to be a passive farming jungler you can. however, i doubt your playing jayce jungle to be passive, that sounds boring! jayce ganks will never last very long, you go in, full combo and leave, if they survive, let your laner deal with them or finish them off quickly, but staying for more than 20 seconds is just a waste of time

some of you may have noticed how i recommend hexflash, truthfully, its not a neccesary rune, but it gives you alot of early presence since you can always just hexflash over a wall to engange.
Ive already mentioned how flexable jayce is but he really can build most AD items very easily so you can use runes to either fill in weaknesses from your planned build OR you can use your runes to amplify your items tenfold. I tend to take a damgage focused primary rune and use my 2ndary runes to fill in for weaknesses.
also note, cosmic insight + iononian boots decreases your flash cooldown by 65 seconds and your smite cooldown by 20 seconds!

Standard Bruiser - Conq is a really good runepage for jayce, he can activate it fairly quickly and he does really well in extended fights. I take triumph over PoM because in the jungle you can always auto camps to get mana back with your W passive. Alacrity is really good for clearing your jungle, while haste can allow for better ganks, jayce prefers to gank early game and then farm midgame while rotating to good fights. last stand allows for you to get out of sticky situations you should be in, which if you play jayce, your gonna q into some bad situations
Assasin- You might have noticed for assasin i opt for cash back instead of hexflash, this is because yoaur taking relentless hunter anyways and this rune page requires strong snowball. You take atkspd shard for your clear because you wont have alacrity, and I believe that flat health shard is much better so you can win the early skirmishes this rune page wants. Dark harvest is so you can take an early snowball really far with early stacks. Sudden impact is so you can drop as much damage as possible before you get out. Grisly momentos allows for you to take sweeper and make sure the enemys map stays dark the whole game. You want relenless so you can run around the map making as much imapct as you can everywhere.
Gank runes - Phase rush is already a pretty good rune on normal jayce, but for jungle jayce it changes your playstyle in a very *odd* way. This rune page wants you to go in, drop damage, and leave, if yyou dont kill them chasing might be possible, but you are often better off just leaving to go farm or gank another lane. Nimbus cloak means your short cooldown summs (cooldown boots + cosmic insight) give you a burst of movespeed, which means your clear because much faster when you smite your camps. Celerity makes all this movespeed even better, celerity takes this runepage and bumps it up 7%. Waterwalking allows for you to have extra AD in those river figths you want to force and gives you even more movespeed going down river for ganks.
Green Bruiser- its just normal bruiser runes but with extra tankiness, its fine, i prefer the ultilty to the extra tankiness, but to each their own.

Ionion boots are best boots, the give cooldown and summoner haste, and summoner haste allows for some really useful tricks.
To be completely honest, I think jayce can build almost any item as long as it contains AD. Of course axiom arc would be worthless, ive even seen crit jayce work wonders, at the end of the day, build ad items and you'll be fine

Health Bruiser- your stacking health and ad, i firmly believe tank jayce may *exist* but its far worse than stacking HP bruiser items. This build is amazing if your team needs a frontline because if gives you tankiness without sacrificing all of your damage.
AD bruiser- a much more standard jayce build, where you build more damage rather than HP, you still heave good survivabilty and sustain with eclipse shield and sundered sky healing, but you have less health, overall a really good build for most games.
Assasin- this is your snowball page, if you get an early lead or need to end a game before the enemy can scale, this runepage will help you take over and win out quickly, you sacrifice all of your survivabilty for dmg and you will fall of late, throwing in a hubris is nice, but only if you are REALLY fed