The CoreCLR has both a MIT License and a patent promise so it should be ok to use it to build your "competing" CLR.
The patent promise isn't unlimited but:
Microsoft Corporation and its affiliates ("Microsoft") promise not to assert
any .NET Patents against you for making, using, selling, offering for sale,
importing, or distributing Covered Code, as part of either a .NET Runtime or
as part of any application designed to run on a .NET Runtime.
and there are some limitations about what a .NET Runtime can be (your implementation must include all the features needed in a .NET Runtime to make it compatible with ECMA-335... so you can add pieces but you can't remove some pieces).
Note that the DNX is probably more similar to what you need: it uses CoreCLR and includes a compiler and other pieces. It is:
The DNX (a .NET Execution Environment) contains the code required to bootstrap and run an application, including the compilation system, SDK tools, and the native CLR hosts.
The DNX is released under the Apache license. The Apache license includes a patent grant in the license.