Sure, you can sequence some of your own DNA at home. It takes a more than a basic chemistry set, but if you have some skill in the lab you'll be able to do it. (You don't though, do you? You're a troll who haunts "Religion and Spirituality," which is why you asked a question that couldn't be answered by specifying "on the cheap.")
You'll need a centrifuge, a vortex mixer, a couple of micropipettes, a temperature-controlled warming plate or bath, and a gel electrophoresis apparatus. If you're handy you can make everything but the micropipettes yourself to save money. You'll also need lab supplies (3 mL test tubes, pipette tips, polyacrylamide gel, phosphate buffer), a PCR kit (detergent, protease, nuclease, polymerase, primers, deoxynucleotides, buffer), a supply of restriction enzymes, and some sequencing kits (polymerase, deoxynucleotides, dideoxynucleotides). (The sequencing kits will come with dyes, but since you're doing this manually at home you can do without them.)
Harvest some cells, break them open with detergent and mechanical agitation, spin down the cell fragments and reserve the supernatant, destroy the RNA with nuclease then the nuclease with protease. Add primer, nucleotides, and polymerase, alternately incubate and cool for 30-40 cycles. Prepare as at least as many samples as you have restriction enzymes (more than 1 per enzyme is prudent), add one or more restriction enzymes to each sample and incubate. Separate each sample into at least 4 parts (again, it's prudent to have duplicates of each part). Add polymerase, deoxynucleotides, and a unique dideoxynucleotide (and dye, if you like) to each part. Incubate and cool each for 30-40 cycles. Prepare polyacrylamide gels with 1 of these parts per lane (unless you used dye) and separate the fragments electrophoretically. Dye the fragments (if they aren't already marked) and read the nucleotide sequence from the gel. Once you have the sequences for all fragments, use the fact that the restriction enzymes cut at different sites to reconstruct the original genome from the sequences of overlapping fragments created by cutting at different restriction sites.
A PCR kit will cost about $75+/-, a Sanger sequencing kit good for 20 runs is about $100. The micropipettes are the most expensive item if you build the rest of the kit yourself.