I have a webapi that is designed to process reports in a queue fashion. The steps the application takes are as follows:
IMHO, your ASP.NET Web API application shouldn't run those background tasks by itself. It should only be responsible to receive the request, stick it inside the queue (as you indicated) and return the response indicating the success or failure of the received message. You can use variety of messaging systems such as RabbitMQ for this approach.
As for the notification, you have a few options. You can have an endpoint which the client can check whether the processing is completed or not. Alternatively, you could provide a streaming API endpoint which your client can subscribe to. This way, the client doesn't have to poll the server; the server can notify the clients which are connected. ASP.NET Web API has a great way of doing this. The following blog post explains how:
You can also consider SignalR for this type of server to client notifications.
The reason why background tasks are hard for ASP.NET Web API application is that you're responsible to keep the AppDomain alive. This is a hassle especially when you are hosting under IIS. The following blog posts explains really good what I mean: