I am not the best at the game, and am trying to improve. But to me, solo Draven is ridiculous to deal with when playing a melee. If I play a mage mid I get bursted down quickly, if I play a tank top I get denied so much farm. Can I get some help?
Play a champion that has sustain, a strong gap closer and/ or who can cs well under tower. 
comes to mind, for the sake of example Ill use him. Basically he is going to push because he has to keep his axes up. Let him push lane and farm under tower until you get one item. After that you will out scale him for a short time. When he next auto attacks charge him and taunt while he moves to get his axe. Best case scenerio he will miss his axe and lose a ton of dps and you will outtrade him completely. Worst case scenerio he will E you and catch his axe getting a few free autos off. If that happens, don't panic just pull back a use your lifesteal to cs and wait for another opportunity.
Secondly the fact that your lane is always pushed should result in jungler ganks. If he is pushing alert your jungle. You can't expect them to gank if you don't show them that you are paying attention. If they don't gank, don't sweat it, it's not worth raging over.
There are plenty of other champs and strategies that can work to beat him as well, but this is one of the safest choices. As an added bonus Shen is a great champ to use to learn how to optimize even the smallest advantage.
well, most melees can't do anything on their own vs a good adc(hence you nearly always see adc in competetive 1v1 turnaments), though vs draven you can eventually use his bouncing axes to your advantage, as you know where he will be to catch them, so you can often force him to drop an axe or eat an skillshot ADCs toplane are really vulnerable vs ganks, as you can easily force out their escape befor your jungler appears, and they are so squishy that towerdives are rather easie.
