Presuming the C shape is a closed path on top of the rectangle, you can just select both and Pathfinder>Minus Front, then Ungroup if needed.
That will cut the overlapping part of the C shape out of the rectangle.
In older versions it would be Pathfinder>Subtract from shape area, and you might need to press Alt/Option to get simple paths, depending on version.
Hi Jacob Bugge
Thanks for your reply! I finally had the time to test it.
I must say it is already more how I would like it, but now the thing is: the C-shape has to remain white when I place the logo on for instance a red background. But only the C-shape within the blue rectangle, so the parts of the shape that come over the borders of the blue rectangle, should be gone (like the effect of a clipping mask actually... )
Then what about:
1) Select both rectangle and Cand Pathfinder>Divide;
2) Delete unwanted parts (of the C), you may Ungroup or use Direct selection Tool?
You can still change the rest of the C to transparent by deleting those parts too, or by hiding them (in the Layers palette, or setting them them to nostroke/nofill), if your client changes mind again. In the latter case, you can (re)show/fill them if your client changes mind again again.