The casino credit hosts can assist you with this. Basically it is the same process as applying for a credit card. You can also deposit a cheque or a cash deposit at the cashier's cage and draw against it at the tables. These three methods will usually garner you more comp credibility than coming across the table with cash. Because you have to sign a marker which involves the floorman, your play will also be more accurately reported.
Casinos also have a hotel room rate that they call casino rate.
This is a room charge lower than normal for people that play in their casino but perhaps do not qualify for full comps. Most comp rates are calculated on the basis of a three to four night stay and cover two people. In general, the cheaper the hotel you stay at, the more comps you should get for your money. You will get free drinks and cigarettes while you are playing and may be able to sign for drinks at the hotel's bars as long as you don't order anything too exotic. You may also be able to sign for meals in the buffet or coffee shop and drinks at the bar.