使出吃奶的劲儿It's a very informal expression.
It's like to say an infant does all it possibly can to get its mother's milk, sucking at the breast as hard as it can. It's really not easy for a little baby to do such things, so this expression is kind of desperate.
So it means......strain every muscle, use every ounce of one's strength.
With给我(for me), it's like an order, an imposition.
Soldier: Sir, it's like mission impossible, please......
Officer: No way!!! You have to make it!!! 给我使出吃奶的劲儿来!!!