The first thing to do is isolate the hole (or holes). The hole will typically be a centimetre or two below the waterline, so run your finger around the liner, just below the waterline, until you come across the puncture. If you can't find it like this then start working upwards.
If the puncture is quite large (ie, more of a rip) then you may need to apply a piece of spare PVC liner over the affected area. If it's just a small puncture then you just need a small length of repair tape to fix it.
I bought a 1m length of PVC repair tape from my local garden centre, and it cost £3. I only used a 10cm length, so now have 90cm tape left which I'll probably never use. Contact me if you'd like me to send you a piece!
Cleanliness is the golden rule before applying the tape. Wash off the worst of the pond slime that has probably built up on the liner, then use a mild abrasive (I used a very fine sandpaper) to properly clean at least 7cm of liner around the puncture. Use paper towels to dry the cleaned area, and be careful to keep mess and pond water off.
You can now apply the repair tape. Cut the correct length that you need, and push it hard over the damaged area. Knead the edges of the repair tape into the liner, so that the edges thin out and form a good bond. Wait for ten minutes, then carefully peel off the protective paper backing. You should be able to fill the pond up straight away.
The whole procedure is very much like repairing a bicycle puncture; you just need to make sure that you're using the correct repair material!
If you found these instructions helpful (or if you had trouble with them) please let me know!