I've never used paint.net, but Gimp (another absolutely free program) can do it:
Open your logo in gimp.
First you need to add an alpha channel, this adds the eventual transparency. to do that:
To your right, you will see the Layers window, and your picture will be the only layer. Right click on it, and select "Add Alpha Channel". Nothing will change, but you can now get rid of the white. Grab the magic wand, and click on the white spot, then hit Delete. No more white. Repeat as needed. When you are ready to save, you click File > Export. Hope this helps.
Gimp:
GIMP - The GNU Image Manipulation Program
-Eddy