Your client is a person, too. They have been late or missed deadlines at some point in their life as well. Unless you have a really aggressive client, they should understand that things happen. And if they are extremely upset about it.. they might not be the type of client you wan to keep around.
I take a very relaxed approach to communicating with clients. Not to say I take my time, but rather I just talk to them like I would any other person.
"Hi XYZ, I'm having some issues with ABC and I won't be able to make the deadline. I'm researching the issue now, but what's the absolutely end date we can push the deadline to?
I apologize for the delay but this was an unexpected issue that popped up. Hopefully I can figure this out sooner than later."
Of course you should communicate in your own voice.. but I don't see a problem with just being straight forward with them. I know I appreciate honestly a whole lot more than I do someone who keeps me in the dark.