You need to open the pores first so hold your face over a bowl/sink of hot water with a towel over your head for five minutes to let the steam open up your pores. Then use an exfoliator or a toothbrush to clean the blackheads (be very gentle when using these!)
Alternatively, you can use a deep cleansing mud mask or black head strips. Both work well just as long as you steam your face first.
Never try to squeeze out blackheads with your fingernails as you risk scarring your skin!
After you're done, apply an astringent such as witch hazel, a toner or either cucumber, lemon, orange or lime juice to close the pores.
Unfortunately there is no permanent solution (that I know off, barring surgery) to getting rid of blackheads, so you'll have to do this regularly.