Generally if you keep the sun behind you, you should be able to get great pictures of the Colosseum from anywhere. If you are looking for specific spots that serious photographers would recommend, and if you are willing to make some effort to get there, then pick one of these places:
Colle Oppio or Oppio Hill
This is one of the famed seven hills of Rome. It is a great spot to photograph the Colosseum. Position yourself right above the Colosseo metro station or up Colle Celio opposite the Platino and Colosseum exit.
Via dei Fori Imperiali
You also get great views from Via dei Fori Imperiali. If you can, go on a Sunday because the Via dei Fori Imperiali is closed to all traffic until 4:00 pm.
Piazza del Colosseo
Piazza del Colosseo, close to Via Sacra....in between the metro stop and the Arch of Constantine is another excellent spot.
If you are around the Roman Forum, get a shot from the front of Roman Forum keeping the Arch of Constantine to your right.