This is very advanced stuff. I'm seeing some ink drops, which are often practical effects (you drop dark ink into water and film it against a white background, e.g.). In addition, it looks like they are using 3D characters in conjunction with some particle effects. If it were me, I'd have a character animating, then use certain parts of the character's surface to emit "smoke" particles, and configure those particles to have an ink-in-water effect. But, again, this is advanced stuff.