Why up to emerald?/Why without Zhonya?
In lower elo you can easily play without Zhonya, you're just exchanging survivability for damage which is good option when you know how to position yourself especially before and after ulting. In high elo you will still have to play with good positioning that's why I recommend you to play without Zhonya at first.

GENERAL

Neeko relies on an aggressive playstyle for maximum damage. Try to turn into a minion as often as possible, EVEN if your opponents know about it (just remember to reduce your speed). Disturb enemies over and over, use W for escape/block.

E > Q = maximum damage, (E must hit another target first, e.g. a minion, to upgrade it)

once you get R, try to use it on opponents with 60-70% HP or less, always with an ally. YOU MUST BE A MINION!

combo: R > Flash > Hextech > E > Q > W
Hit as many people as you can with E! In order to help allies approach in time.

You can skip Flash/Hextech if you can get there safely without being noticed or use W instead, but it's more risky.


MID/LATE GAME

Look for team fights and skirmishes where you can use R on at least 2 enemies (or 1 if necessary). Your engage into battle is VERY STRONG, BUT you must have the help of allies, so wait for them!

Remember to use disguise as much as possible, opponents will forget about you.
ALWAYS use R with DISGUISE (because they don't see it for 0,5s of 1,25s cast time)

If you have 3 or more items completed you can even annihilate some opponents, use it wisely to kill the backline while catching everyone in enemy team with R.



little tip:
If you have no where to run, use W and CHANGE DIRECTION OF MOVING but let clone go where you were going. (IT DOESN'T WORK UNDER ENEMY TOWER, because you're not disappearing)
This way, you can go even towards enemy tower, try to execute, hide in bush or just waste as much time of enemies as possible.
Remember as a support in general your time is less valuable than laners, because they can lose CS.