You need a razor blade, a piece of sandpaper to remove the mold release (don't use the cheap cheese graters that come with some poor patch kits), vulcanizing glue and the patch.
Firstly, remove the old patch. If it's a glueless patch this should be easy. If it's glue type patch, see "removing patches" here: https://www.sheldonbrown.com/brandt/patching.html
Secondly, cut the seam away near the place where you put the patch using the razor blade, carefully to not worsen the puncture. The tube should be completely seamless in the area where you intend to patch.
Thirdly, remove the mold release from the to-be-patched area from the tube using the sandpaper. Make the sanded area slightly larger than the area where you put the patch.
Fourthly, spread put vulcanizing glue with your finger around the area where you put the patch, making it slightly larger than the patch. Wait for it to slightly dry so it looks like a gel and not a liquid. Close the glue tube so that you push the glue at the tip by compressing the tube so that there will be no air bubble and then while compressing the glue tube to keep the air bubble removed, put the cap on.
Fifthly, remove the aluminum foil behind the patch so you have a patch on the transparent plastic only with no aluminum.
Sixthly, press the patch on top of the area you want to patch.
Seventhly, hold the edges of the patch and spread the patch and the tube to see if there's perforations that cause the plastic on top of the patch to spread into two pieces. If you don't see an opening crack in the middle of the plastic, you need to make the perforations yourself with the razor blade. Make a short half centimeter long cut in the middle of the plastic, carefully to not puncture the patch. Then spread the patch and the tube together. You should see that the cut spreads from the middle to the edges of the transparent plastic. You have now a patch on top of which there are two rectangular pieces of transparent plastic. Remove them starting from the middle of the patch.
It is important to note there doing the seventh step correctly is necessary. If you try to remove the transparent plastic from the side, the patch WILL lift. It will cause a poorly patched tube with a slow leak.
Now that you have a patched tube, put it away and hope you don't need to use it in the next 24 hours. It takes 24 hours for the glue to fully dry. The initial short wait was not enough for that. If you try to use the tube in the 24 hours, or if you do the seventh step incorrectly, you will have a slow leak. However if you carry only one spare tube and have to patch in the field, a very unlucky day with two punctures means there's no other option than to risk having a slow leak by using the patched tube in the 24 hours after patching.
The best patch kits (Rema Tip Top) used to have a perforation in the middle of the transparent plastic. Probably to save cost, it has been removed today at least in Rema Tip Top kits so you need to create it using the razor blade.