This one is from way back. When I was in grade school we had a bunch of old Macintosh computers (and probably some old DOS machines as well). This was pre iMac era equipment. Hopefully that gives you an idea of the time period.

I remember watching kids in my class playing a game on one of these old machines. The game was turn based. The background was black. The game was from a 2d side on perspective (like donkey Kong, or super Mario bros) and your enemy is somewhere off screen. You lob munitions from your base toward your enemy. I think you might have to hunt for your enemy with your shots like battleship. You start off with small stuff like hand grenades and as the game progresses you get more powerful weapons (all the way up to nuclear weapons). You take one shot, and your enemy is allowed to fire back, and so on.

The thing that I remember most about the game is that your health is represented by an avatar (similar to doom guy), but as you take damage the flesh is progressively stripped from the face of the avatar revealing a metal skull with red eyes underneath (kinda like a t800 from terminator).

That's all I can remember. Any help would be great.