Many of the characters you will start working with will unlock thanks to the Legend Log. Any remaining characters will require you to encounter them and then beat them in the Dream Log scenarios. By the time I had conquered Legend Log and started on the 21st island in Dream Log, I had only conquered about three strong enemies in DL and had 10 characters left to unlock.
Dream Log unlocks:
Ace
Kuma
Magellan
Moria
Mihawk
Doflamingo
Whitebeard
Blackbeard
Fujitora
Garp
Kizaru
Akainu
Eneru
Kuzan
Shanks
All other characters are unlocked through completing Legend Log, so do not worry if you complete Legend and find that you still have a few locked portraits. For me, I had to complete another island AFTER finishing the final tough enemy in Dream Log.