You don't have to use layers but that is going to be the easiest way.
Pay close attention because this sounds difficult. It isn't though, it's just a little hard to explain. Find a suitable Logo
Go to Google and find a VT logo. Google image search for "Virginia tech logo." You will get a lot of choices. Pick one that has a white background. For instance, this one: Image: http://collegefootballbelt.com/Logos/Virginia%20Tech%20logo.gif Paste in New Layer
Now right click on the image and copy. Go to your Mario image in Paint.net. "Choose Paste in New Layer." It will likely ask if you want to resize the canvas. Yes to fit the logo. You can resize later.
Now you have 2 layers which should be shown in your "layers tab." Make sure you are on the logo tab. Use the Magic Wand tool and click on the white area. It should highlight everything outside the logo. Click delete. This will delete the highlighted section. It should leave you with just the logo. (to make it look best, you want to apply a feather effect to clean the jagged edges). Resize the logo.
Click on anywhere with the arrow selection tool. The whole layer should be selected. Now grab one of the corners with mouse, hold tab and move the mouse in to shrink the image. Tab makes sure it resizes without changing the aspect ratio (fancy way of saying without distorting the height vs width properties).
When the logo is the size you want, use the selection arrow again to move it on the hat.
Now, you have been working in the second layer. So your original mario layer should be unchanged. When you save the image in PNG or JPEG format, it will flatten your layers and you will no longer be able to edit them individually.
It sounds like a lot but this is pretty easy when you get the hang of all the tools. If I was unclear, please ask.