What you are asking to do is extremely dangerous. I have a blog post that explains why this is a bad idea, including code for a BackgroundTaskManager type that minimizes the chance that something will go horribly wrong. Unfortunately, even when you register your background work with the ASP.NET runtime, it's still a bad idea to depend on this behavior.
The proper solution is to have your WebAPI request place a request into a reliable queue (such as Azure Queue), and have an independent service (such as an Azure Worker Role) process that queue.
