The best method of approach is very personal.
Some have mailed a letter so as to lessen the shock and give the receiver time to process before responding. One thing to consider, though, is the possibility of someone else fielding the letter and the privacy of the searched person being compromised.
Many choose to call, but consider again, the person’s household may not know about the adoption, so be discrete if someone else answers. If you are able to get an email address, that may resolve both concerns.
Here’s one suggestion for how you might approach first contact:
"Hi, My name is (......) I'm sorry to bother you, I have kind of an unusual question, do you have a minute? I was adopted in (location) in (month & year) through (agency) and I'm interested in connecting with my birth family. My search has lead me to you as a person who may potentially have information regarding my adoption. Are you able to help me?"
Practice the initial contact with a friend and discuss possible questions or things to say. This may be difficult, but remember you have support.