#1 That mildew may be coming from behind the wall where mildew is growing.
#2 New caulking will not stick to old, so all of the old stuff needs to be removed first, most often a single edged razer and a Scotchbrite pad will get it off. Then clean the area with bleach and water, let dry and recaulk.
Fill the bathtub 1/2 full of water before caulking, let dry 24 hours then drain the water. It will settle the tub so when you do fill it again it will not strech the caulking and crack it.