I main the Carbon in competitive play. You must get good at FAST lateral movement and QUICKLY closing the gap. It's all about SPEED and SURPRISE.
As with any main, you need to leverage its best feature, and Carbon has the fastest OHK in the game. So practice evading and moving forward until you're great at it. You also need to know range really well. Not just the Carbon's range, but all the weapons in the game. You should always know exactly how close you need to be and how to get there without getting splatted. When in doubt, sneak.
As a Carbon, you can beat any other roller because of its speed. Timing the gap in other weapons' rate of fire and attacking in between is the meta of Carbon. It takes a lot of practice and I'm still learning how to do it. Also, being aware of chip damage helps because a weakened enemy effectively extends the range of the Carbon's OHK. And it can roll over weakened enemies! I think that's the theory behind the ink storm special. But in practice it rarely works. You just have to be aware of weakened enemies and change your approach accordingly.
As for team play and support, I'm still learning. In theory, the Carbon is a flanking/melee weapon. I'm having some success at that. And the ink storm is great for map control and pushing an objective. You'll need to learn to adjust really well because the Carbon isn't really a very flexible weapon, and team work requires you to be flexible as possible.
As for essential abilities, I recommend swim speed up, bomb defense, ink recovery, and ink resistance.