Characters are unlocked in both Mission mode as well as the Story mode. They are fairly linear but I'll list them just to be clear.
Uryu: Story Mission 1
Grimmjow: Story Mission 3
Kenpachi: Story Missions 2-3
Nnoitra: Story Mission 4
Rukia: Story Missions 3-4
Kyoraku: Story Mission 6
Soi Fon: Story Mission 6
Toshiro Hitsugaya: Mission 6
Hollowfied Ichigo: Story Missions 5-6
Ulqiorra: Story Missions 6-7
Halibel: Story Mission 8
Barragan: Story Mission 9
Stark: Story Mission 10
Byakuya: Story Missions 8-10
Yoruichi: Story Mission 11
Ichimaru Gin: Story Mission 13
Aizen : Complete Storymode
Kokuto: Mission mode 5
Skullclad Ichigo: Mission Mode 20
Ichigo Final: Mission Mode 23
That should be all of them. If you're missing one I haven't covered, feel free to comment and I'll gladly add them in!