Yes, your .NET Core application can run on Docker! This isn't the right place to provide step-by-step instructions, but we can point you to the right places.
The very best thing you could do is rush right out and buy a ticket to DevNation/Summit next week. We'll have several Red Hatters giving talks about both .NET and Docker, you'd surely hear everything you need to know there. (Our own Don Schenck is an especially qualified SME in this area.) But if you need an answer today, you might start with Google and watch for great articles from Microsoft's Scott Hanselman and others. If you don't mind using Azure with this mix, there's a nice write-up at .
Here (Simplifying ASP.NET applications on OpenShift with the ASP.NET Core S2I Builder – Red Hat Developer Blog) is an example using Red Hat's OpenShift.
I will post a link to a video and blog post by July 12 (2016).
But in short: Yes, .NET works well in containers.