So, haven't tried this before, but here is my guess at the approach.
You've got four points in screen space, in 2D. Let's refer to those as the UL,UR,LL,LR corner points (upper left, upper right, etc.).
Create a vector from the LL to the UL (so, along the left edge), and another vector from the LL to the LR (along the bottom edge). The cross product of these two vectors (with normalization applied where needed) give you the normal of the plane for the checkerboard.
You've still got to figure out the distance of the plane from the camera; to do this, you need to figure out how the edge distances compare with what you believe they should be, and so back out the perspective scaling that's been undergone. To get a feel for the scales, you'll probably need a calibration phase for your code where you place the checkerboard facing the camera straight on (perpendicular to the camera) and move it to different (measured and known!) distances, observing the detected lengths of the edges. From that information you can figure out the perspective transform of the camera.
That should be a good start, anyways.
I made a mistake here and failed to address that the actual depth of the points/vectors would still need to be worked out. The cross-product as spec'd above would be useless, as it could only give you a vector perp. to the camera, which is useless.
I'm leaving this up in case it spurs any ideas, but the approach I quickly sketched is, at best, deeply flawed.