How do you upgrade 5 amps socket to 13 amps socket?

It does not alow you to extend your ring main but your are allowed to replace single sockets with doubles, and you are allowed to add spurs. You may only add a spur with the same number of sockets as the 'host' on the ring main that you are running it from. In your case if you simply add a new socket alongside the existing one by breaking into the ring and fitting it then that would be 'extending'.

If you feed your new double outlet by means of a short wire from the existing double outlet then you have created a 'spur'. Which is allowed. The simplest way to do it is to buy a double socket and raised surface mounting plastic 'patress' box.

Get a length of 2.5mm flat twin+earth cable and a roll of green/yellow earth sleeving. Isolate the ring main. Unscrew the existing socket and identify the Live, Neutal and Earth terminals.

There may be two wired Earth terminals, that's ok. The socket will probably be mounted on a zinc plated box sunk into the wall. Do not remove any of these wires.

Open one of the spare blanked holes in the box and carefully bore or chisel a tunnel from that hole through the plaster to a location on the wall surface about 2" away. Feed your new wire though the new bore and connect it to the back of the existing socket. Make sure that you don't disturb the existing wires.

Slip a length of green/yellow sleve over the bare copper cconductor.