Jax needs to focus on early farm and utilizing his burst in lane so that he can dominate his lane, pick up kills and farm, and snowball out of control. Popular itemization is Blade of the Ruined King + Trinity Force (Usually going for Phage or Sheen first) followed by several tank items.
Your marks should be either AD or hybrid penetration marks depending on whether you want to bully with AAs or bully with abilities and item procs.
Post 6 a nice trick I've done (Albeit with AP Jax for the lulz) is to hit minions twice, then Q (leap strike) to an enemy and promptly layer your W (Empower) with the passive for your ult (Grandmaster's Might 100 MD proc). With either AP or a Sheen item this will do astonishing damage, sometimes chunking off as much as 400 damage in that small exchange. After that you would react to your opponent's reaction (Re-react?).
If they try to fight back, continue slugging away at them by attacking and activating your E (Counterstrike) to minimize the damage they can do back to you and follow up with a stun. With good timing, you'll get a second R proc and possibly a second Empower off for even more damage.
If they run, either try for some more damage if you're sure you won't get ganked, or go back to farming while your opponent tries to heal back that large chunk of damage. If they come back to lane, do it again.
You're not a tanky fighter outside of your ult, so you must have items and bulk. To get those items you must have gold, so again focus on maximizing farm and lamppost slap your opponent if they try to get farm while you can spare the attention. When you get your Sheen-based item you start hitting a huge power spike due to the combo outlined above, but certain champs like Renekton or Xin Zhao can still trade effectively with you due to inbuilt sustain and CC that you don't have as much of. Tailor your next item to resist their primary source of damage. (Xin Zhao is a particular problem as he fights similarly to you, leaping in with a gap-closer and applying auto-attack modifiers and/or Sheen Procs while also reducing your armor by 15%)
An option that you can take is to simply spec into heavy AD/AP against full AD/AP opponents, while relying on your ultimate and counterstrike for damage mitigation, making you more of an assassin that does huge damage.
Another option to consider is how your opponent's team is specialized. If they have heavy MD burst, you could use an AP build in conjunction with MR items like Spirit Visage, Banshee's, Merc Treads, etc. You will have more ability burst, and your ultimate will tack on much more MR that allows you to wall skillshots and damage for your team (One time I hit something like 350-370 MR with my ult active)
On the other hand, an AD heavy enemy team would necessitate more physical resistance. In that case BORK works for Attack Speed and sustain, Trinity Force gives you the extra burst you need, and the rest of your items can focus on Armor (Randuin's, Frozen Heart, etc.) to reduce the AD scaling abilities and auto attacks of the enemy ADC.
) then building tank items like banshees veil and randuins ome. You won't do the most damage, but you will still do a lot and your team will have a tanky guy. Its not idea but when you gotta do it you gotta do it.
