How to upgrade existing ASP.NET application to ASP.NET vNext
ASP.NET 5 (vNext) is currently in beta, so you're not going to find much documentation as a lot of the available libraries are still changing. Likewise, the Kestrel HTTP server, for cross-platform hosting is also currently under development. You can download the release candidate of Visual Studio 2015 here and play around with it yourself, but don't expect any official upgrade documentation until it's release.
I believe there won't be a direct upgrade option. The architecture diverges so much it makes it impractical to 'upgrade' existing project into vNext.
It is easier if you rewrite your application instead.