How can I create a face for extrusion with a hole inside in edit mode? In my case I want to extrude only the "C" from a mesh marked in blue and not the filled innside part.
I'm not sure what you basic mesh looks like, so it's hard to tell what you're supposed to do, I guess you can create edges between the existing vertices but I recomposed your shape with several successive extrusions along the letter profile. To make an angle round I've selected an edge and use the bevel tool (ctrlB). When you have finished with the flat mesh, select all and extrude up (E). Is it what you want?
Edit: And oh yes, as Duarte explains, if you have the same amount of vertices on the internal and external edges, you can simply select them and press W > Bridge Edge Loops. If they don't have the same amount, create enough vertices to make the count equal. You can play with the bridge parameters on the bottom of the Tools panel, on the left of the 3D View.