HI Peter, with your years of experience contacting lost cousins, how would you approach word a message to a DNA contact that probably doesn't even know of you or your parentage? It could come as a great shock to find out they have an unknown relative.
I'm assuming that in this case you're talking about a very close relative - closer than 2nd cousin - since it wouldn't be a surprise, let alone a shock, to discover someone more distant.
Whenever you're communicating with someone always try to look at what you're writing from their point of view. Why would they want to hear from you? Why would they want to reply to you? How can they help you? How can you help them?
Obviously you'll want to take into account whether they're researching their family tree, and if so how long they've been doing it. Clearly they only tested their DNA recently, otherwise the match would have popped up earlier, so you might well be the first person to contact them.
Consider all the possible reasons why they might have taken a DNA test - are they looking for answers, or was it just a 'fun' birthday present. Either way, they didn't test because they wanted to get bombarded with questions!
Also consider the implied morality of the supposed relationship - might they feel that you are in some way disparaging them or their family? Try to present it the other way round so that, if anything, it's your ancestor who is perceived as the naughty boy/girl. If you can't do that, don't mention it at all.
Make sure you tell them enough about yourself that they'll feel comfortable replying to a stranger. Never allow the other person to get the idea that you're chasing an inheritance or trying to right a wrong.