When you do something viciously, you do it in a cruel or brutal way. A bully might viciously shove a smaller child, knocking him to the ground.
You might have seen someone delight in crushing a bug under his shoe — you can say that he does it viciously. You might also describe your angry little brother as viciously pummeling you as you pass by, if he's furiously hitting and kicking. Viciously comes from the adjective vicious, which originally meant "of the nature of vice, or wicked." The Latin root, vitiosus, means "faulty, defective, or corrupt."