Flying to Bari is definitely the best option if you're trying to reach Dubrovnik or the surrounding area.

Bari - Dubrovnik by Ferry
Jadrolinija operates an overnight ferry on this route once a week during off-season, and several times a week during summer.

C/F Azzurra Line also operate the Bari - Dubrovnik route, from the end of April to the beginning of September. The operate a ferry about once a week during this period (sometimes not quite, so check their timetable carefully!) but have a very frequent service in August.

Flights from the UK to Bari
Ryanair have flights from London Stansted and Dublin

Useful Travel Links
Bari Airport Includes information on getting to/from the airport
Trenitalia Train timetables for journeys within Italy