To get from Eilat to Ein Bokek (Dead Sea) and then from Ein Bokek to Tel Aviv, you can on regular buses. I recommend tourists to travel in tourist coaches for security reasons.

From Eilat there are group transfers through the Dead Sea on Mondays, Wednesdays and Sundays. From the Dead Sea, you can travel to Tel Aviv as part of the group almost any day except Fridays.

You need to order such transfers through travel agencies and it's better to plan your route in advance, since the number of transfers is reduced in the low season.

You can rent a car and drive around Israel by car.

From Eilat to Ein Bokek bus number 444 bus company Eged. The bus runs 4 times a day. The first flight leaves at 07:30.