How to approach agents, without becoming annoying or pushy
In the first instance I check out their website and see how they like to be approached - more often than not it's a case of sending an initial email or letter saying you'd like to be considered for representation and if possible invite them to a show you're in. Including links to your online CV and showreel is usually standard (every agency has their own preferences), though I wouldn't attach any huge files to an email, as it can make it difficult to download and isn't necessary if you have a Spotlight/CCP page.
I try to keep the tone formal but friendly - this is always the difficult balance, as you want to come across as yourself but also without seeming to be trying too hard! The way I see it is, you're writing to a fellow professional and should engage with them as you'd want to be engaged yourself. The more you know about them and who they represent, the better, and an initial email doesn't need to be anything more than a few lines asking them to take a look at your details, invite them to a show, and why you think they'd be the right agent for you - this is always enough to open up an initial dialogue, and you can take things from there if they're interested.
Of course, more often than not it'll simply be a polite rejection explaining why it is they can't see the show or why they can't take you on at present. The main thing to remember - and this is always the difficult bit! - is that it shouldn't be taken personally. Just make a note of their reply and try again at a later date. If you remain polite and focused like this it can only be a good thing in the long run, because if you're seen to be working consistently and handling yourself with respect and professionalism then an opportunity is more than likely to play into your hands at some point.
The only other experience I've had with writing to agents is to those who represent friends of mine - and while this is handy in that it can establish a connection and familiarity a lot more quickly, I've known people who have been far too gushing and scared the agent off, because they assume they can let their barriers down! To me the process isn't any different - you still want to handle yourself with dignity and professionalism! It's great when you can use these connections to your advantage, but in the excitement of potentially gaining interest from someone I have to force myself to remember that, at the end of the day, you're talking to someone who you should see as a professional equal.