The cheapest way I know is to get a bus from Nazareth all the way to Amman using a company called Nazarene Tours. The bus departs almost daily at 8AM and will cost around $20. You will be dropped off around Amman University. A public bus from Tel Aviv to Nazareth will cost you less than $10.
Another company called Mazada Tours has a direct bus from Tel Aviv but the cost is around $90. Not worth it.
If you don't want to take a private bus for some odd reason, you can take a public bus to Bet She'an (around $15), a taxi to the border ($15), then a taxi from the border to Amman ($55+).