Dortmund Airport is a relatively small airport. The only direct flights to Dortmund from the UK are with Ryanair.  

The airport is located about 6 miles east of Dortmund. You can get into central Dortmund in around 22 minutes with the Airport Express bus, which goes to Dortmund main station (Hauptbahnhof / Hbf). This service is run by the airport, so the fare is not included in any public transport multi-journey passes. Details on the Dortmund Airport website.

If you're going straight to the stadium (called 'BVB Stadium Dortmund' for the Euros), the quickest route is via the Airport Shuttle bus to Holzwickede station and from there with the RB59 regional train to the stadium's own station. The shuttle bus is also run by the airport, so the fare for that is also not included in any multi-journey public transport pass you might have (such as Interrail, the national Deutschlandticket or regional WelcomeCard Ruhr). Details of the shuttle bus service can be found on the Dortmund Airport website and the full journey can be planned on the local VRR public transport site.

If you're planning to pick up a hire car from the airport, the better-known rental companies based there include Avis, Budget, Europcar and Sixt. You can see which offer the best prices here at Rentalcars.co.uk.