Hello BadCerebral!
I've experimented with this sort of stuff recently, so it's definitely possible. Scan it into your PC and save it as a .PNG file. You'll have to crop and make sure the image has the right borders you want to appear in the game. Once you add the graphic as a 'tiled background' in Construct, you can re-use it as much as you want (drag and drop it onto the layout) and rotate, scale, etc to suit your needs in the game.
You will have to use some collision masks (like an invisible solid sprite) to make sure the pieces act as platforms. One easy way is to put all the invisible collision masks on a separate layer and make that layer invisible in the game so all the player sees is your artwork.
I love the style, dude; there's something unique about hand-drawn images in a game.