How can business jets use slot coordinated airports?
Depends on the particular airport, but I will take The Netherlands (Which has three slot-coördinated airports, including EHAM, EHRD and EHEH) as an example here. 

According to the ACNL all flights to and from a coordinated airport can only be executed when an air carrier or any other aircraft operator has been allocated an airport slot. According to Article 2, paragraph g, of the Regulation as state flights, emergency landings and humanitarian flights are exempted. General aviation is therefore subject to slot coordination (at EHAM and EHRD). However, according to the LOC GA AMS, helicopter and GA propeller-driven aircraft under 5700kg MTOW are exempted from the slot system.

In conclusion: Regulations depends on country and airport. You should carefully read them before starting your flight to and/or from a coordinated airport.

A detailed PDF on how to request a slot can be found here.

GA slots for Europe can be requested through e-Airportslots