Let us assume that white wins. Then, every position White passed through should receive positive credit, while every position Black passed through should receive negative credit. If you iterate this reasoning, whenever you have a set of moves making up a game, you should add some amount of score to all states from the victor and have to remove some amount of score from all states from the loser. You can perform this for a bunch of computer vs. computer games.
Now you are having a data set made up of a bunch of checker positions and respective scores. You can estimate the features over those positions and train your favorite regressor, such as LMS.