Ideas
Soldier:
Should be as it is. Seems perfect in the demo. Bullrushing and just stabbing a guy...just perfect. Aggressive yet somehow sort of elegant.
Infiltrator:
Be based on stealth and martial arts. I can see neck-snapping, throat-slitting, stuff like that. Various sorts of martial arts takedowns leading to knife/omni-blades to the face/chest.
Vanguard:
Even more aggressive than the Soldier's. Maybe something like a Biotic Punch? Mini-charge, a haymaker with the force of biotics behind it perhaps. Basically, even more aggressive and brutal than the Soldier's. It'd probably be too much to put in, BUT some X-Ray effects to show the crushing of bones a la the new Mortal Kombat would be epic and brutal.
Adept:
Maybe something like fashioning a blunt object out of a mass effect field to smash someone? Kind of like a mass effect baseball bat, or something where you just create that field and wield it as a tool of destruction. More finesse than the Biotic Punch of the Vanguard, but it has its own style.
Engineer:
A punch with the omni-tool, but the omni-tool discharges an incredibly strong electric current. Thunderpunch (15/15PP, lulz) for the win, eh? I could also see it being a randomized "elemental" effect: one time you have the electroshock effect to represent Overload, another a freezing effect to represent Cryo, and as already mentioned, a fire effect to represent Incinerate.
Sentinel:
A biotic punch with the omni-tool. When the punch connects, the omni-tool emits/detonates a charge unleashing an explosion or mini-shockwave similar to how Assault Armor detonated in ME2. The enemy receives the tech explosion, then the raw force of the biotic force that supplements the tech effect will summarily end the opponent.