"How" do I exercise a Call option?

Say, now it's May and XYZ company's share price is $50, and I bought its Oct $52 (striking price) Call option at $2.5 (premium). Two months later, if the share price goes up to $57 and I decide to exercise it, I suppose (from many investment books and online knowledge) I will have $2.5 profit per share (i.e., $57 - ($52 strike + $2.5 premium) ). However, I've seen demos on several brokerages' websites (as you can tell I'm new to this) and I'm really confused... By the time I try to exercise a Call option (not as I bought), after choosing 'sell to close' as an action, I will be also asked to enter a 'premium' again; what if the premium has also gone up (say $5 for XYZ companys Oct $52), what premium should I enter (again, upong exercising, not buying)? What's the significance of that premium (that is, what role does it play when I exercise my option)? Because of it, will I profit more? or will I lose part of my profit? Can anybody answer my question? Many thanks!