I have a code written in C# I would like to use as the back-end of a site I'm building.
I would prefer not to build the site front-end in ASP.NET (which integrates nicely with C#), and to use PHP or Python instead. 
Is that reasonable? Should I re-consider using ASP.NET?
How can I achieve that?
Just use asp.net mvc framework for the frontend instead of plain asp.net. It's easy to learn. And if you know php it will be easy to you undestand asp.net mvc.
I don't see the reasons if you are using c# backend  use php frontend. For sure you can create service layer on c# and communicate with php through it, but it does not make sence for me.
Connecting C# Back-end with PHP frontend