Re: how many points do you think we'll need from these 6 games?
It's not as simple as points, because two of the matches are against direct competition for fourth.
Coming into that penultimate weekend where the Villans visit Eastlands, expect Villa to be in fourth, Liverpool just behind them in fifth, and City a couple of points back in sixth.
Villa should be favored in their next four league matches (Everton, @Portsmouth, @Hull, Brum), so I would expect eight points to be the minimum haul for them from those four, with 10 or 12 points being a very distinct possibility. And with Blackburn at home in the final weekend, Villa should get at least 11 points aside from their match at Eastlands.
Liverpool should win their next three (Fulham, West Ham, @Burnley), figure to lose to Chelsea, and win at Hull on the final match day for 12 points.
Spurs don't figure to get a single point in the next three matches (Arsenal, Chelsea, @United), but they should bank three at home against Bolton and at home in the finale at Burnley. So that's six, aside from the match at City.
So ignoring the City v. Villa and City v. Spurs matches, that gives us:
Liverpool 67
Villa 65
Spurs 64
I would say the most likely outcomes for City are win over Brum, draw with United, loss at Arsenal, and win at West Ham. That's seven points, putting City at 66.
So if City can win at home over Brum and at West Ham, and get a result at home against United, then they can draw with both Villa and Spurs and still finish fourth.
These results in other matches, combined with a win over one of Villa or Spurs and a loss to the other should still mean fourth for City.
A loss to Villa and dropping of points against Spurs figures to see Villa finish fourth, while a draw with Villa and a loss to Spurs should see City and Liverpool finish even on points.
So, to answer your question, I would say City need a minimum of nine points from their six remaining fixtures, but the key is not losing to Villa.