The way I like to play Jax early game is to not harass until you reach 2 or 3 or basically until you have your E. With your E you're able to Q to someone E and get away unscathed. You can wait to level 3 when you have W so you can get a bit more damage in but mainly wait till E. Once you hit 6 you don't have to last hit as much because your passive and ULT passive go good together by just hitting them and when your ULT passive triggers the minion should die. If you're facing squishy opponents that you know you can kill then wait till you hit 3 and poke em down then at the right moment engage. Late game you're good if you did ok early game don't ever engage first you're better off being a damage source NOT an engager. Rather then that there isn't much more I can tell you so I hope this guide helps you and you're able to do good in solo-que with Jax.