1: Play something that can deal AoE damage. Malz was my choice, but VelKoz might be the deal now.
2: Master the shit out of said AOE champs (2-3 at most).
3: Play ANTI-engage. Only make plays if you are CERTAIN there is NO ONE there to stop you. Read: Map awareness like hell. Anti engage means staying close enough to do something but far enough back not to get mashed by dives. Say your team is in jungle and their team dives in after yours... as Malz/Vel/Someone AoE/ you launch everything and wrekt some face down as they are clustered. Enjoy as you either wreck their faces or your team cleans up.
4: Do not try to make desperate plays--only if you're VERY certain it will get away clean. That means NO FUCKING TOWER DIVING WHEN THEY ARE FULL HP WITH ALL COOLDOWNS UP! :O~~
5: Play anti-engage. Because your teams are going to be hit or miss in reliability when you do engage. Refer to #3 and #4.
6: Don't go past river without TONS OF SIGHT AND YOU HAVE ADVANTAGE.
7: Don't chase.
8: See 3&4. And 5.
I see you have Azir play. He's long distance enough with high enough damage / ult that makes a difference that might work out.
Being "jack of all trades" of champs isn't really going to work out well. Playing all champs to get used to their ranges, weaknesses, CDs, etc. is excellent but not to play them all the time.