I have a room at the back of my house where there is very poor/no wifi signal.
There is only one phone/ethernet port at the other end of the house.
What do I need to make everything betterer?
(I don’t want to run a cable and would rather not get a new port installed).

Something like this;
http://www.netgear.com/home/products/networking/wifi-range-extenders/EX6150.aspx

I have 2 of these which cover upstairs and our extension out the back of the house. Seem to work ok for me.

I used this one to extend my wifi from my house to the shed so I can watch Netflix on the turbo. Distance is about 15m and works a treat

Wifi Extender

I got one of these from Sainsbury’s at christmas time to see if it would allow me to get wifi in the garage at the side of our house.

TP Link TL-WA860RE

Works great and has the plug through function that my be important if you are short on sockets.