I am trying to mount a 20 inch TV on a plasterboard wall. There are only two studs in the wall. One where I won't be able to see the TV and the other above an electrical socket. I do not want to put the bracket on the stud with the socket. Nothing is in line in my house, so god only knows where the cable is. The only other wall is a chimney breast. I am reluctant to mount it there, in case I get it wrong. Also my drill is a bit week when going through brick. I really want to do this my self and not ask my brother. Many thanks.
Check out "togglers". They work extremely well. Two of them should be more than enough for a 20" TV. Along the togglers you'll need a 1/2" drill bit.
I use "AUGER" style screw in anchors if i have a heavy load on sheetrock with no stud in the area. There are several types plastic and metal the metal ones are better (IMHO), and if renting they can be removed and the hole filled. The metal ones many times can be reused , the plastic ones are usually toast when removed. if you google "auger sheetrock anchors" there are even videos on how to use them hope this helps.
While mounting to studs would be preferable, one strategy I've seen done involves getting a sheet of plywood larger than the mount and mounting this to the wall - giving you the ability to use more anchors/screws than the TV mount allows. Perhaps adding an adhesive like this would increase the surface-contact area. You can then mount the TV mount to the plywood. With the glue, keep in mind that your wall will be trashed behind the board if you ever decide to remove the plywood. The purpose of this is to spread out the weight of the mount+TV over a larger area.
Then add some decorative trim and/or paint to match.
Regarding drilling in brick, make sure you're using a (not dull) masonry bit. That may explain why you're having trouble drilling through it. There are also relatively inexpensive TV mounts on MonoPrice made specifically for mounting on brick/concrete (example, not endorsement).