The role of .net native in .net ecosystem is confusing for me. I heard it is just for universal windows applications, but also heard that it is part of CoreFX. I think having the option to compile to .net native can have many advantages (including performance).
Is it possible to compile my asp.net application (specially asp.net 5) to .net native?
No, you cannot. Right now, .NET Native is not for ASP.NET. I believe it's only for Universal Windows Applications. That doesn't mean that one day it won't be available, but right now it's not planned.
See related GitHub issue where ASP.NET team confirms this.
Since this was posted, Microsoft has made further announcements regarding .NET Native and .NET Core. I suggest you check out Scott Hanselman's part of the keynote from the Microsoft Connect 2015 event. At the 11:22 minute mark of this excerpt video Scott shows compiling an .NET app to native code and then running it. He says it's "future work" so it appears it's not quite ready yet (I believe one of the Q&A videos from the event explained that it's in one of the dev branches on GitHub, but I'm too lazy to rewatch all the videos for you at the moment). It was unclear if this is only working for console apps at the moment or if it will run ASP.NET.