It would certainly be an interesting experience. Recompiling an entire Linux distro using an alternative compiler would not really be straightforward however. Many essential packages depend on GCC and other GNU toolchain utilities.
It would be feasible to use an alternative compiler for those packages that don't require GCC to create a hybrid. It might be possible to replace GCC-dependent packages with others that would build with your compiler, but in most cases there aren't too many alternatives. Where there are, you'll have to determine whether those alternatives will be as functional for your intended use.
The real question though is: would it be worth it? If you are trying to build an embedded system, an alternative compiler might produce smaller binaries. If you are building a compute cluster, you may get faster code execution. If you are building a desktop, you'll get a great learning experience.
I think in the end (assuming you are building a desktop) you'll find to be true what many source distro users eventually decide: the amount of time spent optimizing and building and rebuilding far outweighs the time saved by faster code execution. That is not to say that it shouldn't be done -- it could be quite fun. If you do it, this site would be a great place to find answers and document your efforts.