Generally you should use a patch kit similar to this , which consists of patches with rounded corners & feathered edges, a small tube of glue, and a bit of coarse sandpaper.
Usually you can locate the leak by over-inflating the tube (to about twice normal diameter) and then either moving your hand along it or moving it around near your face (your face skin is quite sensitive to air movement) to feel the leak. Sometimes it helps to move it around near your ear to hear the leak. As a last resort, fill a basin with water and use that to detect the bubbles.
Discard any tube with a leak within two inches of the stem, or with a leak within an inch or two of an existing patch. And of course discard any tube with a slit or blowout (except in extreme emergency).
Marking is tricky. Often the leak is near enough to the center of the tread area that you can simply center on the tread area and only need to mark the position around the circumference. Other times the leak is more on the sidewall, and you need to mark both X and Y (so to speak) positions. (Or you can wing it and try to remember where the leak is.)
But remember that you need to abrade the area of the patch (plus at least a half inch beyond its margins) and then spread glue on this area. This will obliterate any nearby marks, so you need to mark farther away than that. It's often best to mark on both sides, so you can conceptually draw a line between the marks.
Chalk, a soft pencil, or a fine-tipped felt pen can be used for marking. (Be sure to test marking first and be sure you can see the mark.) Avoid using a grease pen, since the grease may get onto the patch area.
Abrade the area of the patch. If the patch will fall on top of the mold sprue (seam) on the centerline of the tube, you need to abrade that pretty well -- not necessarily enough to remove the "seam", but enough to make it lay flat.
Spread on the glue, going about 1/2" beyond the patch margins. You don't need a lot, but enough to make the whole area shiny and smooth. (Note, you WILL get glue on your fingers, especially since the finger is the best way to spread the glue. Be prepared with some sort of towel to wipe your hands, and don't do any of this on your wife's new hardwood diningroom table, nor should you wipe your hands on the upholstery of the new couch.)
Allow the glue to dry. Here I may diverge from "conventional wisdom" and the instructions in the patch kit, since I like the glue to be not quite totally dry before applying the patch. Others like it thoroughly dry, but I find I get better adhesion my way.
Apply the patch. It comes with a clear plastic top cover and a foil backing. The trick is to remove the foil without removing the clear plastic, so that you can use the plastic to cover some of the glue while pressing. After the patch is in place, press it thoroughly, by, eg, rubbing your fingernail back and forth across it. (The smooth, rounded end of a plastic tire "iron" is useful for this function.) It's important to do this pressing well, crossing the patch first one direction, then another, applying considerable pressure.
If the patch is on the sidewall where you will fold it while folding up the tube for stowage, allow the patch to "set" for at least another 15 minutes, preferably a couple of hours. And, before folding for stowage, place a bit of tissue over the glued area (even if the clear plastic is still covering it) since the glue, over time, will want to stick to parts of the tube it touches when folded. (Another option is to thoroughly dust the area with talc.) You need the same consideration when installing the tube back in the tire, to prevent the patch from sticking to the inside of the tire. (In a pinch, when you need to put the tube back in a tire and ride immediately, roadside dust is a good stand-in for talc.)