I am thinking that you probably have tar and gravel on your flat roof.
If it is ten or more years old. It would be a good idea to do a tear off and check to see if any or all of the sub-roof needs replacing.
Since you are asking for advice on how to repair it yourself, I am assuming that you don't want to hire a roofing company.
If your roof is tar and gravel, you can scrape it off with a flat edged shovel, (it is a lot of work), but once you have the old top portion off, check to see if any of the plywood sub-roof needs replacing. (You'll need a janitors broom to sweep the sub-roof off with, to clear any loose bits of gravel off). If you see any deterioration in the sub-roof, replace those pieces, then you can lay felt paper, but you will need to overlap the felt at least 8 inches per line. Make sure to over lap the same where each roll ends. You will need to pop some lines in order to keep your paper straight, so pick up a chalk line and powder.
You will need a bucket of Simplex nails, (small nails with round plastic heads on them).
Once you have the tar paper,(felt), down, nail the simplex over all the edge lines of the felt where you have overlapped the paper, about two inches apart, with the simplex nails. Including the end of roll overlaps.
You will then need to lay down some bull pitch, (black tar). I would use the (wet or dry plastic type). You'll need several buckets, depending on the size of your roof. I recommend; (gardner- Wet-R-Dri All Weather Plastic Roof Cement.
Thourougly apply the bull over the tar paper in strips as you lay the rolled roofing over top of the paper. Making sure to cover the simplex nails really well. Do NOT use tar/pitch sparingly and make sure it is spread well on each edge of the felt paper.
You can use a paint roller with a long painters handle on it to roll the tar onto the roof.
You'll need a razor knife and plenty of sharp blades to trim the edges with along the edge of the roof line. Wait for the tar to set up before trimming the edges.
Total cost in materials will depend on the size of the roof.
It is difficult to find a leak on a flat roof and try to patch it with tar, but if you feel the task of replacing the kitchen roof is too much, and you can't afford a roofer, than you could try to locate and patch it with tar/bull roof cement.
You can pick up everything you need to get the job done, at a local supply store such as; The Home Depot.