> But I just want to know: What is the relationship between Philosophy
> and Science?

That's not a simple matter. F.e. nowadays some metaphysicians, like Ladyman and Ross in *Every Thing Must Go*, argue that metaphysics can and should be informed by physics (to some degree). 

> Personally, I think the scientific method must be assumed as an
> axiom/premise in Philosophy.

There's no such thing as *the* scientific method. In detail, the various conceptions of what the scientific method is supposed to be do vary.

>  My argument is that there is no way to deny the sure success of the
> sciences (like Physics, Chemistry, Medicine, etc).

Success isn't the same as truth. That is, we can hold scientific anti-realism and think that the results of our scientific processes are empirically adequate (see Van Fraassen) or functional *while* agreeing that the enterprise of science is important and successful.

> So, what is the relationship between Philosophy and Science?

Like others mentioned, many philosophers think that science rests on a backdrop of epistemology etc. But some philosophers think that it can go the other way around. Some moderately so - I've given an example above -, fewer formulate it quite radically. Take for example Quine who thinks that epistemology and metaphysics can be "naturalized", that is, explained by natural science. For epistemology he thinks that the explanation can be done with psychology etc. Although he sort of argues against a distinction between science and philosophy in general. Quine is heavily criticized (and imo rightly so), so keep that in mind.

edit: typo