Biotic Ascension possible in single player?

no, but you can kind of build for it by choosing a profile which reduces shield cost, and choose skills which refill shields.

eg: vanguard with lance, pull, and shield drain or charge.

also, mod of adrenaline resets your skills on a kill, so that might also help you get close to a biotic ascension type character.

Personally, I would choose sentinel as a profile, as he has shield cost reduction, shield restoration and combo damage as bonuses, and also has the damage reduction through tech armor bonus.
Then use shield powered lance, a biotic primer ability (pull, primer throw, primer shockwave, singularity maybe?), and primer shield drain.
This creates a constant loop of combos, shield usage and shield restoration.