You don't need to be a chopper to win with backspin, even in this modern age of topspin. Backspin will always have its place, when used properly. Of course, even attackers often serve backspin, hoping for a pushed return (backspin) that they can loop. However, when should an attacking player use backspin in a rally?
Backspin Against Topspin: Chopping (which is backspin against an incoming topspin) can be tricky, especially with a fast attacking racket, and so usually an attacker will use backspin only against an incoming backspin, i.e. a push. There is one main exception. If you are forced off the table, and attack from the wide forehand, an opponent may quick-block to your backhand. Ideally, you can attack this as well, either backhand hitting or looping. However, if you are out of position and have to reach for the ball, a better answer might be a backhand chop. This keeps you in the rally, gives you time to get back into the point, and it may throw off the opponent's timing.
Short Pushes: If you push short (so the ball would bounce twice on the opponent's side of the table if given the chance), then your opponent can't loop it. If you keep it low, it's very difficult to attack effectively, and will usually result in a pushed return that you can attack. Key to this is keeping the ball very low to the net. Don't just dink the ball back; brush it back lightly with backspin, which will make it easier to keep the ball low and short. Learn to change directions at the last second, dropping the ball short to the middle, left, or right. Short pushes are especially effective if you push both long and short, so the opponent never knows what's coming.
Long Pushes: There are many ways an attacker can win with long pushes. The key to long pushes is to push quick, low, heavy, deep, and at wide angles (or to the middle against some two-winged players). You should also often aim one way, and at the last second change directions. (So aim to the backhand and go quick to the forehand, or vice versa.) Here are some other ways to win with long pushes: