Pathfinder enables regular multiclassing (taking levels in multiple classes) as well as variant multiclassing (substituting feat slots for class levels). I was thinking of making a character who does both of these, taking multiple levels in different classes and spending feat slots for variant multiclassing. Is this possible?
It is; variant multiclassing uses up your feats, and does not care what your “actual” class levels are doing. The one caveat is that you cannot take actual levels in the class you choose for variant multiclassing. Paizo recommends against using both, however. As with any variant system, you should speak to your GM about whether or not you can use it, and whether or not you can use it the way you want to.
It is probably a good idea to use either this variant system or normal multiclassing, but it's possible for the two systems to be used together. In a game using both systems, a character can't take levels in the secondary class she gains from this variant.
Beyond this suggestion, allow me to also offer my own recommendation: never use variant multiclassing at all. Not a single one of them is even remotely worth what you can get with the five feats that you give up. Most of them are insultingly poor, in fact.