As with any professional correspondence, you don't want to seem too stiff or formal, like you're writing a letter from a Victorian template or a bad spam email asking someone to send money to a deposed prince overseas—but you also don't want to be too conversational. You're not in a dialogue yet, so it's important to treat this like a professional first interaction, and not like you're skipping several steps and asking to meet for coffee. The tone you're reaching for is, "You don't know me yet, but I'm interested in your company and want to tell you more about why."
Without job-level specifics, it can be tempting to get stuck in an "I'm awesome" loop without giving enough specifics. If you have a general idea of what job groove you'd be seeking at this particular company, build your letter body around that. If you're truly just trying to get a foot in the door at a company, you can use clues from your research about the company. What does their website say about their mission and priorities? What do current and past employees say about the company in online chatter? Even without specifics, you can cobble together a pretty good idea of what the company is seeking in potential employees. Sell your skills and experience points that are special to you, and especially relevant to your industry.