I am not sure how you made the rain but there are many ways to achieve a depth look...
In the past, I got pretty good results using CC rain (cycore filters).
The key is to put the effect on an empty comp inside your main comp rather than the footage layer (in your case the logo). Using an empty comp will create an alpha channel for the rain and you can have several 3d comp layers offset in Z-space (in front of and behind your logo).
All you need is a Camera to get DOF.
Another possibilities would be Trapcode Particular.