If can often be best to find cheap flights and book your accommodation separately, using this website to find a hotel:
http://www.hostelworld.com/
Don't be put off by the word 'hostel' in its name; you can use the filters to select only hotels or B&Bs; I found a lovely little hotel in Madrid on that website, in a superb location, with air-conditioning and free wi-fi for just €22 per night for a single room in peak season.
A quick look at that website suggests that you can find good accommodation in Dubrovnik for around £13 pppn if you don't mind sharing a bathroom or for just a little more for en suite facilities.
Unfortunately there aren't very many scheduled flights from the UK to Dubrovnik though. The only year-round service is that operated by BA from Gatwick but there are also seasonal services as follows:
from Edinburgh, Gatwick and Stansted with easyJet ;
from Birmingham with FlyBe ;
from Belfast International, East Midlands, Edinburgh, Leeds/Bradford, Manchester and Newcastle with Jet2 ;
from Birmingham and Gatwick with Monarch ;
from Birmingham, Bristol, Gatwick, Glasgow, Manchester and Newcastle with Thomson.
Transfers from the airport to your accommodation aren't expensive. The bus into the city only cost 35kn (€5) or there are reasonably-priced taxis.