How to contact casting directors
I think a great start is to research into the casting directors c.v. most are found on the casting directors guild. It gives you a good idea on the type of shows/plays they cast for. Obviously it's most relevant for what they have recently or are currently casting. I normally acknowledge it very briefly in a sentence or two.
I generally email quite a short friendly and specific email with a reason to contact them. I.E. to introduce yourself because I have a new showreel or perhaps a new agent or set of photos and I would be appreciative of any feedback and/or consideration for current projects.
An invite to an upcoming play or film showing is perfect but remember their diaries get booked up so if it's specifically for an event then I suggest 6-8 weeks notice and a follow up call/email. Tho it's a fine line between being a pest and being an actor sometimes. Try and stay on the actor line. Difficult and frustrating as it is.
The more work you do the more reason to be in touch with a casting director and I'm finding they eventually remember you and hopefully this helps build a relationship.
As a lot of Casting directors are t.v. or film then a showreel is perfect. A short showreel with good stuff is worth it's weight. No fillers. It seems from most casting directors that a montage isn't necessary. Simple is best.
As for the email/letter, nothing to long or flowery just polite and brief with the relevant link.
If you are inviting them to a show I think a follow up email or possibly a call would be fine.
I would try to avoid blanket emails tho I am guessing most casting directors wouldn't really mind it is just more professional to make the email personal and friendly. If you come across a bit vague and abrupt then to be fair you probably are not going to appear much fun to work with.
Finally, it's useful to contact casting directors with the rest of the actors in the production. If they are not initially interested in you well who knows you could catch their attention off the back of another actor. Acting is a team on stage/screen and also off.