Integration of ASP.NET with the maps generated by MapTiler is equal to any other web technology.
MapTiler prepares for you a directory structure with hundreds of small JPEG/PNG tiles and a sample ready-to-use HTML/JavaScript viewer which allows zooming on these tiles. If you upload all the files with directories to your web server, same as you do with normal images, then the viewer will display the map online in your websites.
The next step is then enhancement of the generated sample Google Maps API viewer with ASP.NET - to add binding to your database or other dynamic functionality.
This is the easiest way.
If you have large number of maps or if you want to avoid to transfer thousands of small files to the server, then you can choose to render the MBTiles format on this MapTiler screen:
In such case the whole map is saved in a single file which is easier to upload to the server, but you need to use a simple tileserver to serve the tiles. We have developed one in PHP (https://github.com/klokantech/tileserver-php/) and in C/C++ (http://www.tileserver.com/). I saw online an alternative made in ASP.NET as well: https://github.com/sainsb/tilecannon