Chelsea's goal difference is +22, West Ham's is +11. So the latter will probably have to win all their remaining fixtures and hope Chelsea drops at least 4 points. It's possible, but what looks easy on paper may not be so in reality.
I think it's a race for two spots now. If Leicester should drop points against Newcastle, they're right in the thick of it, and other than that they have remaining games against us, Chelsea and Spurs. Two losses in those four games, and they'll drop below 70 points. They still have a greater margin of error on the others though.
Chelsea is I think still in the best position among the remainder, despite four difficult games remaining (City, Arsenal, Leicester, Villa). They can afford to lose one of them and still reach 70 points, if they win the rest.
West Ham need to win all of their remaining games to reach 70 - although that does not seem undoable (Everton, Southampton, West Brom, Brighton).
Those are the teams that can still reach 70 points. Tottenham and Liverpool can only get a maximum of 68 and 69 points respectively. Which means that to make top 4, they'll need two of the three teams ahead of them to trip up, combined with not dropping any further points themselves. That seems to me unlikely to happen.
Everton can only get to 67, and the loss against Villa effectively ended any real chance for top 4.