Plan your routes in advance of bidding on slots. As the process is competitive by definition, governing bodies will look into several factors, which include the types of routes you're serving. Specific ideals and requirements will vary by airport, but if you're not pitching a profitable route -- for example, if you want a nonstop flight between Washington Reagan and an airport in rural Virginia with little air traffic -- and your competitor is, it's likely the competitor will win the slot.
Contact the airport authority directly if you seek slots at a U.S. airport. For example, if you want to bid on slots at New York's LaGuardia Airport, you'd need to contact the Port Authority of New York and New Jersey. Visit the website of the airport in question to find appropriate authority contact information.
Call Airport Coordination Limited, or the ACL, at +44 (0)20 8564 0600 if you are interested in slots at any UK airport. Unlike in the United States, the ACL -- a single, independent entity -- controls slots for all UK airports.