(Popeye) - avoid Bluto! Note that the first few levels have upper platforms that let you walk off screen and re-appear on the other side, so you can lead him to one side and take advantage of that to dash down and get hearts.
When he's on the far side and the same level as you, he throws stuff at you - it's worth 100 points for each of these you punch, so worth getting the hang of doing that. If you find yourself on the bottom with him dangerously above you, get close enough that he crouches to do his 'slapping' swipes, and use that moment to run up steps in the opposite direction...