I want to get from TLV to Thessaloniki, Greece on 21-24/June.
What is the cheapest way?
You can try a boat, they used to have pretty cheap tickets on ships between Greece and Israel a decade or two ago, but I'm afraid its not as popular as it used to be, so you might not find any.
Sea and air are your only options, unfortunately.
Aegean Airlines currently has direct, return flights on these specific dates for €241.87 (including taxes and luggage). Olympic Air on the contrary is more expensive and flies through Athens. I believe the Aegean price is difficult to beat.
As overland travel, either by hitchhiking or by public transport (bus+train), is not really an option at the moment, flying will probably be the cheapest option.
On kayak, I have seen direct one-way flights from Tel-Aviv to Tessaloniki for 576 NIS (~120 EUR).
Going by ferry (if you find one) will hardly be cheaper. In 2011, there used to be a ferry service between Israel and Cyprus. The posted price is 250 EUR, one-way. From Cyprus, it is possible to take a ferry to the Greek mainland.
It has been a long time since I was in Israel, but 15 years ago, there were a number of charter operators offering holidays in Europe. I would suggest going to the Ben Gurion Airport website, where you can see a full list of flights.