I've been maining jax since I started like 2 years ago. I play him in 99% of my games and I have over like 1000..I think. Maybe. Anyways I'm only silver but i FEEL like I should be around gold. I've been getting matched with high gold/mid plats lately in my norms. I win around...80-90% of my lane matchups against anywhere from teemo, ryze, renek (probably the bad renek players though..), singed, etc and I get first blood 70% of those wins. You can look up masteries/runes I use on lolking. Ign is infatuated.
These are what I start with/how I lane. Might not be for you but idk maybe it'll help.
Start with a long sword, 2 hp 1 mp pot. Get ignite and flash. Start q, then e, then w, then max w, then q, then e. Ofc add a point in ult whenever you can as usual. If you're vs nasus or singed, you could start w first. I usually go all in at level 2 assuming you hit level 2 before them.
-Prior to hitting 2, harass a little. Walk up to them when they're about to last hit and just auto attack.
-Try not to engage until level 2, and if you do hit it before your opp, be aggressive and zone them out. If they stick around and you get a few more autos, you could consider going all in.
-Remember that your e lasts quite a while. Don't reactivate it unless they run/are running. If they're staying around and attacking you, just let your e keep spinning. They'll do no damage while you wack them. Then finally when it stuns, they should be pretty low, q on them and just chase and ignite and if they get low enough and you're confident, flash in and auto them to get them lower and then back off and let ignite finish. If they're not low enough when they run, save your ignite and let them b so you can free farm.
-ALWAYS HAVE A WARD. ALWAYS. On your first back, get one ward if you can afford it. Prioritize items first though. Always ward your bush. Don't go too far past the river entrance without a ward unless you're sure enemy jg is far away.
-Remember to use your trinket. If you're too far away from your turret and you get ganked, try to avoid/juke some skill shots and then jump to a minion. If no minions. put down a trinket, q to it and then flash. Or flash then put trinket and jump.
-Wait for your opponents to use their skills before you use yours. Like ryze at lv 1. A lot of them start w/ their red ball. Wait for him to use it then q on him and wack him until he runs. Then you back off. He's less likely to keep fighting and will just run which gives you maybe 1 or 2 more autos before you back up because his skill would be off cooldown.
ITEMS
-Start a sword, build into cutlass.
-After your cutlass, if you b and you have enough for a sheen and you're near level 6, consider getting it. Once you hit 6 you can take advantage of your ult passive. Hit a minion twice, and Q onto the enemy, which will proc sheen, and then w as soon as you land so you can put down some W damage + sheen proc. Then back off and repeat. 2 hit minion, q onto enemy. If he fights you can e and either keep fighting or just run.
-After that, if you built a sheen, you can choose to get trinity or build your cutlass into a botrk.
-If you didn't build sheen, just rush botrk. I usually don't get boots until botrk and tri are completed.
-ALWAYS HAVE A WARD.
-After tri and botrk, it really depends how deep into the game it is. If you're still laning phase, get boots. If not, build some hp for randuins or spectral for banshees and get boots later.
-GET A WARD
I stress wards because they're mini flashes with a 6 or 7 second cooldown (your q). I don't know how many times it's saved my life from ganks. ALWAYS USE YOUR ULT WHEN YOU FIGHT. ALWAYSSSSSSSSSSSSSS