You can exercise on the HCG diet but you will typically be advised not to exercise strenuously. The original HCG diet plan does not recommend vigorous exercise for you, because some studies show it may slow down your weight loss. Your level of fitness before going on the diet can help guide you in determining what is vigorous exercise for you.
A good rule of thumb while taking HCG diet drops is to listen to your body. When on a 500 calorie diet it can be hard to maintain a regular work out routine. So that is one of the reason's why it is suggested that you don't work out while on the diet. But if you want to work out just make sure that you are not pushing your body over board.
Well think about it, you need calories in order to support your energy levels. If you're following the diet, and exercising, you could cause yourself to get dizzy and exceed your body's energy requirements. I'd recommend just doing your weights to keep your tone, and skip the high energy cardio. Maybe just do 30 min of walking to get the lymph flow and digestion flowing, and do some toning work.