One of few titles that should be capitalized. Avoid abbreviating to DG.
The hyphen is important. Here's an example from a photo caption which shows why:
Stephen Hawking (left) with CERN Director General Rolf Heuer.
The sentence above could be (incorrectly) interpreted to mean that Rolf Heuer is a (military) General, and he is also Director of CERN.
The hyphen renders the compound adjective unambiguous - Heuer is a Director-General.
The plural of Director-General is Directors-General.