Well I thought of a kind of a pseudo-code to figure it out.
First of all, if it covers only black squares, it is obvious that it has to pass through corners of the squares, otherwise it would necessarily pass through white squares.
Any point on a circle satisfies the relation $x^2 + y^2 = R^2$, where $x$ and $y$ are the horizontal and vertical distance of the point from the circle's center respectively.
If you look at a chessboard, you'll realize that the center of the circle you're looking for must be in the middle of a field (not on a corner), because you'll only be able to draw a circle through the corners where the black squares are neighbouring in the direction SW-NE on the left side, and NW-SE on the right side of the circle.
Having the problem set up in this fashion, you can now check whether the corners that are $0.5$ square away, $1.5$ square away, etc. satisfy the above relation.
Taking a quick look at a chessboard with this in mind, I would say that the largest circle you can draw is from the center of a black square passing through the corners of the farther edge of the neighbouring white square, i.e. with $R = \sqrt{5/2}$.