Heads and tails are the two sides of a coin. "Head" is also a slang term for being fellated, and "tail" is a slang term for female genitalia. By "getting head" and "needing tail" the rapper both receives oral sex and desires to fornicate.
"Box" is a crass slang for female genitalia. Gmail is a popular provider of e-mail, which includes an "in-box" feature. The rapper is claiming that he frequently penetrates female genitalia.
"Birds" and by extension, "chickens" are urban slang for kilograms of cocaine. Our rapper here is claiming that buy distributing a kilogram of cocaine, or "crossing the street" with it, he earned eighteen thousand dollars.
A "package" is urban slang for a delivery of drugs. His "whip game proper" means that he knows to use inconspicuous cars at legal speeds to not alert the attention of authorities when delivering said drugs via automobile.
"Dirty Money" is a euphemism for monies obtained through illicit means. Our rapper here is claiming that he is often in possession of this sort of money, and therefore must "wash his hands" of it.
"Wearing the 45" is also referring to the fact that if it is necessary for Jay-Z to return following a disagreement he will be carrying a "four-five" which is slang for a .45 caliber handgun.