I think since your question is more money-related than car-related, it is probably ok in OT.
$5000 is a decent-sized down payment, and I'm sure you'll get a loan of some sort, but the interest rate will mostly be determined by your credit score and history. At 19, your history can't be very long, but I think you'll still get a loan, especially in the current auto market. Also, if you have any trouble getting a loan, credit unions are good places to start, often offering good rates.
As for the process, you should get pre-approved with your bank or credit-union for a loan amount that will cover your hopeful car, plus taxes, fees, etc after your down payment. Then you can go to buy the car knowing that you can get the loan. The dealer may or may not offer a better rate in response. No harm in having them run the numbers if you've already decided to buy the car. If you finance through your bank, you pay the dealer your down payment and usually have them fax the bank (your loan officer can give you the fax number) a sales invoice showing your down-payment and the remaining amount due. The bank pays the dealership the remaining amount due within a day or so, and you can come pick up your car. The dealership may also be able to work a temporary loan with no early payment fee within a certain timeframe, so that you can drive off in your car the day you pay your downpayment and not have the dealer financing "go through the system" as long as your bank comes through with their payment on time. Read the fine print carefully with that, but it worked fine for me.
Capital One and maybe a few other banks can use a "black check" system where you get pre-approved for a loan up to a certain amount, and then you can write a special check to the dealership or private party with any amount up to your pre-approval limit. That's nice and easy, but you may find better rates at another bank.
Lastly, at 19, you've got a lot of big things to save for in life and you'll probably go through a lot of big life changes in the next few years. Think long and hard about tying yourself down to that much debt so soon in your career. If you still live with your folks, make that your priority over a shiny new car.