I want to go on the birth control pill. Should I be worried about dvt?

Worried? Concerned? Ocp's are are known to be associated with dvt. Some are more prone to DVT than others. Smokers should not take ocp's as a a general rule. Anyone with a history of clot or strong family history of clots should be very careful in considering this form of contraception. Have a very frank discussion with your prescribing physician, a pharmacist and research it yourself.
Generally no. Dvt risk is very low. The risk is higher with a family history of thrombophilia ( a predisposition to a clot in the leg or lung). Bear in mind however that even if there was a genetic disorder that increased your risk (like leydig factor v) the chance for having a clinical event is less then 1%. The general quoted risk for a DVT with the low dose pill is 3%, but up to 8-12% with pregnancy.