Mark,
O'Brien is a fairly common surname with a long history which I believe originated before the time when most surnames began. I have a couple of suggestions for you. I'm going to assume that you are searching for your father's surname, but the following should also work whether that is true or not. Please let us know if you have specific things you are looking to get out of your DNA test. Does your DNA ethnicity results show Irish roots?
Since you really don't have any relatives shown on Wikitree, I would first, on Ancestry look at your closest DNA matches, and look at the shared matches when you compare. By doing this with the first 20 or so strongest matches, you can determine a list of potential surname candidates.
Start building up your information here on Wikitree. Add your parents, then grand parents, then great-grandparents, etc. until you connect up to the global tree. Once connected, you can follow these families back often hundreds of years or more. You can also check what you find against the list of potential surnames found on Ancestry.
As Jonathan mentioned, by the time you get back to about 10 generations back, you have as many as 1,024 potential ancestors which each also have as many as 1,024 potential ancestors that came before them, so it's virtually a nightmare trying to do it all at once. Because of this, I follow the "How do you eat an elephant" rule. By tackling this nightmare one family line at a time you will find it's not the nightmare you thought it was.
Hope this helps!
Edited to fix a typo....