First and foremost, I don't recommend patching tubes, because of the likelyhood that the patch will leak. Also, I do love fixing things and not throwing things away, but I should value my time as well. It's just overall better if you have the few extra dollars to buy a new tube. If you're reading this because you haven't patched a tube before, I recommend you don't learn the art. Pick a different project!
There are a number of reasons the patch will leak air. One of the reasons is that the glue may have gotten old. Just because it comes out, doesn't mean it's runny enough. It may be good to save up a bunch of tubes for patching, so you use all the glue at once.
The type of glue you need is “vulcanizing fluid” (contains petrolleum distillates), like the REMA tip top brand. Rubber cement likely won't work.
When patching bicycle tubes, you have to scrub the area where you will apply glue, because the surface of the tube has a coating that the glue does not adhere to well. If you use a piece of old tube for the patch, you also have to scrub the underside of the patch.
If you have a pre-made patch with thinner outside edges, that's better. They have a thin plastic on the outside. Don't remove that until after the patch has dried onto the tube. I remove the silver lining after I've applied glue to the tube.
Use plenty of glue to cover the area on the tube. It should come out from the underside of the patch when you press on it. You can then wipe the excess.
Once the patch is on the tube, keep pressure on it with a vise, or set it on a desk and add something flat on top with weight. After a few hours, carefully lift off the plastic. The plastic will lift off the thin edges of the patch from the tube, so constantly scrape it off the plastic if it does. Pull the plastic parallel to the tube, rather than perpendicular to it. I've read one person's post where he said he just left the plastic attached. Maybe the plastic will come off inside the tire under pressure, since it does not expand? That would be fine.