Most of the rental companies you can rent a car from have offices right at Alicante Airport. You can rent a car right at the airport and drive the rest of your trip. Several companies, such as Record Rent a Car, Sol-Mar and Gold Car, have offices in the center of Alicante itself.
Unfortunately, it's unlikely you'll be able to organize a rental car without a deposit in Alicante. Most companies charge a fairly large amount of money as collateral and can freeze up to 1900 EUR on your card for the duration of the rental.
Cheap car rental in Alicante is possible if you use the services of Drivalia. But their office is two kilometers from the airport, and you can only get there by shuttle.
Order a cheap car rental in Alicante from Firefly.