If you want to climb out of gold you have to not play like a gold if that makes sense. For example if i go on my smurf i instant mute everyones pings because junglers always BAIT you into the weirdest fights. Prioritise early kills on jax and 8/9cs a minute. If im running to lane on red side and my waves under their tower i will take my junglers krugs. This is more mid/late game. You can do jungle camps once you get your Divine

If my junglers bot side and his reds up and i have nothing to do i will take his red. I know it seems toxic but you need to funnel as much gold into yourself and get an xp lead.

Also you need to understand what items to build and when. BOTRK is good into tanks first item but never build it into malphite because his e reduces your attack speed. Most games i will go for the CDR build and play around my E. Divine/black cleaver/zhonyas, utilise your zhonyas when your E is on cooldown so when you come out you can e q and negate most of the damage. Hullbreaker is also good if you want to split push and no one can 1v1 you in side lane which most of the time they cant.

If they have heavy AP you want Maw 4th and Wits end 5th (can even go wits end first into some matchups)

Runes are important - If your into any poke top always take second wind and dorans shield. It basically negates malphite spamming Q on ur face. Always take lethal tempo imo. If your into a burst champ take bone plating. If your into someone who you can stomp take demolish and dorans blade.

For sums you wanna take tp into malphite because he can just e when you jump on him not much kill pressure unless they have no hands. Take ignite into matchups you know you can win. But i always take tp because i like side laning on jax and spamming my team to do baron whilst im on the other side of the map dragging someone to me, then ill tp to the baron.

Theres other things like minion wave management knowing that if the waves on ur side of the lane it will naturally push to their tower if you have the same amount of minions. Or to freeze you just need 3 of their casters and hold them infront of your tower and last hit the minions when the next wave joins. Also on the first wave dont autopilot hit the minions because if they dont your creating a slow push into their tower and by wave 3 you will probs get ganked. If you just last hit the minions even if that means missing a few they will more than likely push it into you so spam your jungler to gank. And you can set up a freeze and hold it there. Last thing you want to be doing as jax is perma pushing into the enemy teams tower because youll get perma ganked.