Would I be crazy to email the director of HR at [big 4 accounting firm] to introduce myself?
I'll finish my accounting degree soon, and am feeling pretty ambitious, so I thought, "What the heck?" Would I seem like a total nut to email the HR director to introduce myself? Heck, what about the CEO.
I found them all on linkedin. It's worth a shot isn't it?

Nope… won’t get you what you want, instead you will be labeled as pesky.
Instead find current employees that you have things in common with like fellow alumni.
And if you don’t have the background these firms are looking for, such as internships, start pounding the payment to make your application more well rounded.