I use Xubuntu, but haven't used Character Builder. I do however use lots of Windows only tools, so I might be able to help.
I would suggest installing VirtualBox: http://www.virtualbox.org
That will enable you to run a VM of Windows on top of Linux which will thus run that code natively without Wine. If you install the guest OS tools, you can even switch between your normal desktop and the VM without using the input escape key.
As far as Wine and Mono (the OSS version of .NET I believe), I don't think all of .NET can be legally included because of patents MS holds. They have said that some of .NET's base is okay to be implemented elsewhere, but not all of it. For me, the VM route is easier anyways. Of course I could be wrong.