There is no one best way. It will always depend on the person you want to meet, who you are, and why you want to meet them.
I will proceed by assuming that you are an attractive woman who wants to make a male friend in the USA, which I could also have deduced from your profile. You have already begun the process when you patiently but discreetly observe him. When you have exhausted your opportunity for observation, for pehaps you have been noticed, or run out of time/patience/clearance, you need to make your move. First, however, define your exit strategy. More about this later.
You must then initiate a greeting that invites conversation, something like, "Hi, I'm Jenny Craig. I have been here a few times before and this is the first time I've seen you." There are countless other opening gambits, but any genuine expression of compatibility, commonality or candor is usually appreciated. One of my favorites in the South is, "Do you think it's hot in here?"
Be sure to smile and maintain eye contact during the greeting. In the USA it is considered ok to extend the right hand for "the shake," but this is optional for women.
I believe it is dangerous to remark on personal attire or physical attributes, or to direct the conversation immediately onto other people. It is better to ask for opinions about something else in which the subject is presumably interested. (*Which you no doubt discovered during your previous period of observation.)
According to the best authorities I have read on pick-up technique, you should not attempt to extend the first conversation beyond a few minutes. As soon as you are comfortable doing so, activate your exit strategy. Apologize for needing to leave, admit how much you are enjoying the encounter, and ask for an exchange of phone numbers.
In the worst case scenario, the minutes would fly by and you might find yourself having to turn down an invitation for a nightcap, which could destroy any chance of a long term relationship.