I'm wondering if its possible to buy say the DLC for Lego Harry Potter Years 5-7 before starting the game fresh on my console and still have the DLC in the game. Will the extra spells, characters etc be available right at the start or should I play/install the game first THEN buy the DLC? And odes this apply to all games in general? 

First play, then DLC.

Depends on the Dlc. Some Dlc is best used as an expansion to extend your time in game, so should be gotten after playing a while. Others add something to begin with, and help when you just start playing.

I don't think the OP is asking about the principles of buying DLC, I think they're asking about whether or not this is actually a thing =P

 

And the answer is depends on the system. Nintendo games typically lock DLC from individual purchase, and you need to do it through the game itself (which has to do with their terrible lack of unified accounts). PlayStation games have access to the PSN store, so you can buy DLC for a game without actually having to own the game itself (and a quick search through their online store shows the game and its DLC for purchase).

 

I have no idea about XBox, but my understanding is that it's basically the same as the way PlayStation does it, although I don't know if they have a web browser store.

 

As for when the DLC shows up, that's entirely dependent on the game. You might have to play through some of it before being able to access certain things.