You have a few choices to get that domain.
First, look into backorder domain services, these companies try to get domains the second they expire if someone is interested in them. Normally the fee is $60 but some are higher (a higher fee does NOT ensure a better chance of getting the domain). Registering with more than one service only improves your odds of getting it. If more than one person wants it from the service that acquired it, then it goes to auction.
Second, with these companies, if no one wants the domain they may still acquire it and put it on their auction block. But, if no one bids on it then the price drops. I've been successful in getting a few good domains for under $10 this way. This only works with the company that registered the domain.
Lastly, and this is most of the time a slim chance, wait for it to expire, then try to register it with your registrar of your choice. While this is normally the least expensive way to go it is the least effective.
How much is the expiring domain worth to you? Is it a holy grail or a "eh, it'll do" domain?
Also, look at what registrar currently has the domain about to expire and contact them, sometimes they'll sell the domain before it goes to a backorder, or they'll keep it if they know someone wants to buy it.