I don't play Jayce a lot, but you should definitely be getting some more survivability.
Maybe go with BT, BC, LW as your only damage items (or some other 3 item variant) and then go with boots and at least two survivability items conditional to the enemy team.
Obviously, if you're getting blown up, build more defense, and if you're doing just fine, build damage. I would say that BT and BC/LW are pretty important on him, and most AD champions for that matter. Muramana could be good on him, but I wouldn't know. If you do get muramana, then perhaps other mana items would be good as well, such as iceborn gauntlet/banshees.
Randuin's is almost always a strong item against AD with the passive, active, armor, and HP.
Bulwark wouldn't be a bad item to pick up if your team is lacking one.
GA is usually a pretty good item, but make sure to get HP to back up the resistances.
Warmog's/Frozen Mallet seem to be pretty popular now (especially warmogs) as they offer a lot of health.
For the most part, you're probably doing everything well enough, you just need to build a little differently so you don't get blown up. You'll do a lot more damage if you can survive another 10 seconds even if you give up a little damage.
As for laning, that will mostly just take practice and getting to know other champions. Nunu is just tough to beat, but he doesn't do as much later on in the games. Kayle, I'm not sure about, haven't really played much against her but I know her early damage is pretty high so watch out for that. Laning as Jayce involved a lot of poking and very little 1v1 dueling.
Remember, you don't have to get kills to do well. If you put your opponent behind and create an advantage for yourself in level/farm/towers then you're doing great.
- GA/Bulwark/Wit's End?
- Boots (mercs probably)