Without a tablet or touch device, your best bet is to use vector graphics. I mean, some pretty incredible drawings have been done using mice and trackballs, but it takes a lot of time and patience to develop the kind of control you need for drawing on such devices. If you're doing raster drawing though, you're better off just spending $40-50 to get an entry level tablet (especially since you won't have pressure sensitivity without it).
But that particular graphic looks like it was done using vector shapes/paths, and something like that can be easily created using a mouse and any graphics program that supports custom shapes/paths (e.g. Photoshop's pen tool).
As far as learning how to draw people, that's something that takes a bit of study, but there are tons of books on figure drawing and drawing human anatomy. A trip to the local library or a quick search on Amazon should turn up plenty. However, you may want to focus on cartoon-specific books, since that's the style you're shooting for, and cartoon/comic-book characters have very different and highly stylized proportions.
Assuming you have that iPad app, I would start by picking up a copy of Inkscape, Gimp, Illustrator, Photoshop, or even Fireworks, and then start copying the different presidents. Try to copy them as accurately as a possible, going through all the different presidents. And when you're done, start from George Washington again and make another set of copies. With each new set, you should get better, and you can compare the different versions to monitor your own progress.
This might seem tedious and uncreative, but this is a basic training technique that most figure-drawing and painting students have to go through. And it's especially helpful when learning how to emulate a particular style.