If you're familiar with both of these games, perhaps the rules are already clear to you. In case they're not, a turn consists of either making a legal clobber move with the checkers on the board or placing a domino on two empty spaces a la domineering. Thus, the player that clobbers with the black checkers is playing dominoes vertically, while the red-or-white checker clobber-player plays dominoes horizontally.
This is enough to make the game states quite difficult to analyze. Often times the game will seem like it's over in one player's favor, but then there's a really sneaky move that changes everything.
My usual strategy is to concentrate on clobbering to open up domineering plays for myself for later that don't allow my opponent to play dominoes. I don't have any better considerations than that; if my opponent catches the drift, I'm generally in trouble!