In one of these days I would like to start with a new project and MVC seems really interesting, but I was wondering if it is possible to upgrade a MVC 5 project to a MVC 6 p
It won't be a direct upgrade path, as a lot of breaking changes were made. You can start playing with MVC 6 now though, no need to wait. You can download the Visual Studio 2015 Preview and get started.
Some of your code will move over, but you'd need to go through and verify every bit of it and fix a lot of errors. It's probably better to start from the ground up with MVC 6 (Microsoft has recommended this in some Q&A on videos I've seen).
Visual Studio 2015 is officially out now, and ASP.NET 5 and MVC 6 are in RC1 status. You can create a new ASP.NET 5 project with MVC 6, then just copy over all the source files. After that, work though the errors one by one until you've solved them all.