Starting early you want to save your mana, once you get e you can start poking the enemy out with e and then an enhanced auto. Neeko's root hitbox is large when going off of minions and lasts longer so try to use that to your advantage if you're getting pushed under turret. Roaming on Neeko early is extremely good as long as your midlaner isn't too ahead, going bot lane is ideal because then you'd have two forms of cc. Don't forget your W gives vision.

When roaming with Neeko you want to disguise as someone else who may have lower health or isn't doing good in the game. Or, if needed, you can disguise as someone extremely fed to scare people off or mess with their vision. Ult in team fights even if you have a chance of dying if it means you can win the fight UNLESS a dragon buff such as elder or baron is what you're fighting for. Readying ults in river bush and hextech rocketbelting out is perfect for those objective fights. Neeko can also bait the enemy team to fight for drags and barons if they have scuttle, she can W over it creating the illusion that the team is doing it.

Stick with the team late game, as a heavy cc champion you don't want to get picked off. Try taking turrets with teammates with your enhanced auto passive to clear them faster. Backline and try and ult pre-emptively then rocket belt into the enemy team so you won't get bursted to death before then. One Neeko ult has the capability to turn entire games.