What you need is acetone, a glass scraper, GE#1 silicone (the #1 has no fungicide in it), a caulking gun, painter's tape (which I forgot and you can forget if you don't mind a messy job, plastic grocery bags to wipe the silicone off your fingers (a trick that keeps your hands and britches clean), and a shop vac.
Start with a clean dry tank. Make sure it's super clean by cleaning it with acetone.
I usually take the easy way out and just remove the old silicone to either side of the leak which is usually a corner (see pic) It is harder to remove the silicone in the very corner but you can get it.
But when I got a look at the silicone in this tank the silicone was terrible around the entire bottom of the tank so I did the whole thing.
When you have scraped off all the silicone you want to replace, vacuum the tank out. Then clean with acetone a second time. Scrape the area again. You will get off the little bits you didn't know you had missed. Vacuum again then acetone again.
At this point I got involved in doing the job and forgot to take pics until I was done. But you run a bead of silicone all around the bottom, dampen your finger and spread the silicone neatly around the area you are working on. The silicone remains workable for a period of time so with this small tank I did it in one fell swoop. Mine isn't neat because I didn't bother with the painter's tape. It's fine with me, but if you're a neat freak use the painter's tape before you silicone.
Let the silicone cure for 24 to 48 hrs then test to see if it's water tight. If it doesn't leak, drain the tank again and to be on the safe side let it sit empty and dry before putting the tank back in place and introducing fish.
I'm afraid I'm not the world's best explainer. I'm better at answering specific questions. If you have any, please ask.