How to quickly diffuse a dye through still water
If you want a homogeneous distribution of dye, you will have to stir the solution. 
If you want to use electrophoresis, you need a charged dye. 

Azo dyes used for hair coloring have colour and charge, for instance Basic Red 51 and Basic Yellow 87:





Of course, heat will increase the diffusion rate but will also disturb the water.