There are two different class sequences you can get, the ones from the soldier in Alvanista Castle, and the ones from the soldier in Euclid Castle.

Fighter - Master the Psion Kick combo

Knight - Know all of the techs (minus combos and the Soul techs)

Master - Have at least 12 equipable combos

Expert - Master all the techs (minus combos and Soul techs)

Champion - Win one round in the Euclid Coliseum

Berserker - Received after fighting 1000 battles

Samurai - For having the Muramasa from Lanzo in Japon/Ninja Village

Combo Master - You need both the Combo Counter item from Lower Morlia and the Combo Command item from Aegis' Workshop (bought for 5,000,000 g)

Druaga Maniac - You need to have some rare items, especially those from the Ishtar and Gilgamesh legend. The items are: Blue Crystal Rod (forest surrounding Japon), Blue Line Shield (Treasure Hunt), Blue Candle (Treasure Hunt), Excalibur (Lower Moria), Golden Armor (Lower Moria), Gold Helm (Treasure Hunt), Green Torch (Treasure Hunt), Hyper Gauntlet (Treasure Hunt), Jet Boots (practically any store in any city), Red Lantern (Treasure Hunt), Red Line Shield (Treasure Hunt), and Silver Mattock (Treasure Hunt).

Fencer - You must have all of Cless' skills and combos (and have used them all at least once), all of Mint's spells, all of Arche's spells, and all of Klarth's summons.

Master Fencer - Same as before, but all your current party members (read: 4) have to be at lv. 99.

Cless gets some stars next to his level number in the status screen.

Star 1 - Master Psion Bolt and Dual Kick

Star 2 - Be able to equip one of the special combos (Lion Claws (end of Ninja Village sidequest), Lion Teeth (beat the Coliseum once; the last optional boss isn't mandatory), or Thor's Wrath (Lower Moria))

Star 3 - Accept the Berserker class from the Euclid soldier

Star 4 - Accept the Samurai class from the Euclid soldier

Star 5 - Accept the Combo Master class from the Euclid soldier

Star 6 - Accept the Druaga Maniac class from the Euclid soldier

Star 7 - Accept the Champion class from the Alvanista soldier

Star 8 - Have Cless at Master level (lv. 99) (the best place to level up is Lower Moria or in the Coliseum)