I have several old receptacles that are 2-prong. There is armored cable running from the breaker panel to each receptacle. What do I need to do in order to install and properly ground new 3-prong receptacles?
The correct answer is that you cannot install a 3 prong grounded outlet without using a 2 conductor with a separate ground conductor, that's three conductors. I know people cheat and put a wire from the ground screw of the outlet to the metal box and hope that the armor case is firmly connected to the box and grounded at the panel. This is not a good alternative to 14 or 12/2WG and can be potentially very dangerous. Example, if you were to have an open neutral, the ground can potentially take over, now you have current on the armor shield, not good. Btw, there is no good way to remove paint from an outlet. Just replace it. In your case, you should be looking at some new wiring.
The only way you can do it is if what you are calling armored cable has enough room in it to fish a ground wire through. If the armored cable is around 3/8 round then the answer is probably no. If the armored cable is 7/8 (the OD for 1/2 aluminum and steel flex) then there will be room.
Fish tapes come in many different lengths and types and if you can use it some places even rent them. What ever you do, get the flattest tape they have.
When ever you use a fish tape make sure you know where you puling to and from. Never pull in a conduit that has live wire and if you pull into a panel that panel will have to be de-energized.
If you can get a fish tape through then you can either attach the wire or pull a string through that you can attach the wire to it.
Remember when you are using a fish tape it means you are holding a ground wire so don't work it around live circuits.
If you don't have a enough room for a ground wire, then a fish tape might be helpful to get new wire through the wall to your box. Check with somebody that knows local code on what kind of wire you can use, whether armored cable, romex or???
The best way to replace old 2 prong receptacles without reworking the wiring is to use a GFCI type receptacle. This gives you all of the safety and protection that current NEC grounding requirements provide albeit with a slightly higher cost per receptacle.
The GFCI must be marked "no equipment ground". You are not allowed to connect any equipment grounds between the GFCI and any of the other receptacles supplied by the GFCI.