I was wondering what is the cheapest, and fastest way to get from Dubrovnik Croatia to Athens Greece?
There are no good direct plane, train, bus, or ferry connections between the two. Croatia, being a non-EU country, does not have a great number of cheap flights, and most of those are from non vacation spots (Germany, Britain, etc.). Probably the best option is a cheap flight to maybe Rome, then another flight into Athens. Maybe cheaper would be a ferry, Ferry to Bari, Italy then to Patras, Greece, But you would be looking at most of two days of travel.
There are cheap airlines operating out of Split, but as poster above points out, most (all?) fly north. There are trains through Zagreb, but I wouldn't characterize them as fast. I've made that train trip a few times, once all the way to Athens --through Serbia and Macedonia-- and did enjoy it.