Are you logged into iCloud on the new phone
Settings > iCloud log in
Is Photo Stream enabled
Settings > iCloud > Photo Stream on
Is your computer set up with iCloud? If so all your photos should have synced to it through Photo Stream.
It is worth noting that Photo stream only holds photos for 30 dats or 1000 photos whichever comes first. It should not be used as a form of back up but instead a syncing tool. Photos synced to your computer from photo stream are stored permanently on your computer until you delete them.