Some diseases will be easier to check for than others. For example, Huntington's Disease [1] is strongly associated with having multiple repeats of a short sequence (CAG) on chromosome 4.

You would start by designing PCR primers to the sequence around this repeat, typically aiming for a couple hundred base pairs total.

Then, you extract your DNA [2], and PCR out that region.

Finally, you run your PCR product on an agar gel, which will separate it by size, with smaller things running faster. If you have a sensitive enough ladder (i.e. a set of DNA pieces with known sizes), you should be able to see measure the size of the product, and from there estimate how many CAG repeats you have.

A different project might be to test whether you are infected with, for instance, H1N1/09 influenza (aka swine flu). Using sequences available online, find a segment of DNA that's unique to the strain you think you might have, collect DNA (the mouth swab may even have enough if you're contagious), then PCR out that segment of DNA. It would be nice to have a negative control as well, such that if you get a product, you know it's not just some other segment of DNA that happens to amplify off the same primers.

[1] http://en.wikipedia.org/wiki/Huntingtons_disease

[2] Kits available, such as http://www.epibio.com/item.asp?ID=270 . Alternately, can do a phenol-chloroform extraction, which would require a centrifuge and access to some nasty chemicals, though might be cheaper per extraction.