If you use the latest version of MapTiler available at http://www.maptiler.com/ it will directly render a sample leaflet.html viewer for you.
Both mercator tiles or raster specific tiles (for images without georeference) are supported. The MapTiler Pro has also an option for specifying custom size of the tiles.
An alternative would be to render Zoomify tiles. Leaflet can display Zoomify tiles as well. Zoomify tiles are JPEG, and do not contain extra transparent part.
See: http://blog.thematicmapping.org/2013/06/showing-zoomify-images-with-leaflet.html and code at https://github.com/turban/Leaflet.Zoomify