Center of Backpackers Tel Aviv
Tel Aviv Israel Center of Backpacker Universehttp://www.hobotraveler.com/ce_centerofbackpackeruniverseisrael.php
Allenby is the street and a great beach in front, dorms for about 10 U.S. and if you are lucky single start at 20-30 Dollars, not a cheapie country.
This is ground zero for Backpackers.
You can also Try Ben Yehuda Street, It has a lot of short term apts for rent, out of season you could find cheap places to share, another idea is to search in craigs list tel aviv