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So you're suggesting that checking the enemy king with the first of your two moves would result in a stalemate?  Interesting, though it seems like that could be unsatisfying, and might cause a lot of draws.

The obvious generalization of the rules of check is that the king is in check if it could be captured before the owner's next move (which in a doublemove variant would often mean within 2 opposing moves), but this has a couple of issues:
1)  It is not always easy to tell when a piece could be captured within 2 moves, making it hard to determine when someone is in 'check'
2)  The force required to checkmate a king that can move twice consecutively is quite substantial (see Betza's commentary on Monster Chess here: http://www.chessvariants.org/d.betza/chessvar/muenster.html )

And thus, various alternate rules proliferate to attempt to solve these problems.

I invented a doublemove variant in high school (which I imagine has been duplicated by many other inventors both before and since) that required the two moves on a turn to be made with different pieces, and also stipulated that the second piece to move could not pass through the square that the first piece started on (thus, no instant revealed attacks).  This has the advantage that a king is in check in any given board position if and only if he would be in check under the FIDE rules, and the doublemove helps only a little bit in escaping check (since the king cannot move twice).  I don't know if that would be considered a 'true' doublemove variant, though, since no indivual piece can move twice during a turn.

My variant also has obvious generalizations to three or more moves per turn that probably play equally well.  I only ever played it once, though.