Jax is my main champ, and I've played over 300 games with him so I would consider myself pretty proficient in playing Jax. For the most part both of these responses are accurate. Especially the one about bringing support with you . I know my Jax record went from about -20 to +30 since I started playing premades and laning with a Taric/Soraka, since Jax is most susceptable to being killed early on. I especially love Taric because his stun/shard can assist you in getting some much needed kills early and get you fed.
When I first started playing, I used Dorans shield with a health pot as my first buy. I would go Dorans shield, ninja tabi, guinsoo, frozen mallet, bloodthirster, atmas in that order. Since then though I've revised my strategy a lot, skipping the Dorans shield and potion (since i mostly lane with support anyway). I still get ninja tabi and guinsoo every game (unless the other team has a lot of CC and I need merc treads). I've recently skipped frozen mallet and replaced it with phantom dancer which is amazing to have. I also like to get guardian angel since the other team LOOOOVES to target Jax. But until you get comfortable with Jax and see how your play style is, Rhino's build is best. You'll replace some items here and there though the more you play. Like I said, I've played like 330 games with Jax and only recently have been buying Phantom Dancer over frozen mallet. It's hard to give a specific build because a lot of what you need for items depends on what champions you're going against.
As far as laning strategy, I tend to hit empower then leap strike, then run away. This gets pretty boring for a while, but it'll keep your health up and slowly reduce theirs. Once you hit about 5 or 6, start walking in front of your minions to get attacked by the enemy minions (try to not get hit by the champs though). Once you dodge and your counterstrike comes up, empower+leap+counter then beat the **** out of them. Waiting for your counterstrike to proc is probably the #1 thing that brought me from a poor Jax player to a good Jax player.
The key to Jax is patience. Leaping and running is annoying to you but it also makes them scared of getting close and may make them go back to base giving you an exp/gold advantage in the lane. Once you rack up a few kills and higher levels and the team fights start coming, wait for your tank to engage, then jump in and start snacking q/w/e. Depending on your items, you're going to start seeing their entire team drop like flies. Make sure you target the lower health champs and the more dangerous ones first, leacing the tank for last.
Hope this helps