In trying my best to answer this question I tried to look through the New York Times archives, which unfortunately can be a bit hit and miss in terms of finding the right articles.
He apparently was a delegate to the National Union Convention in 1866, but the Kentucky caucuses that elected him and others also as a platform stated they "unqualifiedly condemn the President's defection, fully indorese Congress's reconstruction policy, and look to the loyal men of that body for the protection of our lives and property." Whether that is immediately indicative of Breckenridge's views I cannot say.
Beyond that however there is no mention of him politically, only of his works with the Presbyterian Church.
Left with little more then guessing, I suppose it isn't much of a stretch to see himself as a mixture of being a Radical Republican with the restraint of Andrew Johnson when it comes to amnesty; having many members of his family having fought for the Confederacy or having had Confederate sympathies, I doubt that he will be especially severe in the punishments he decides to met out, and like Johnson he will face a lot of ridicule and disdain from Radical Republican from it, to the point where some may claim that he himself has Southern sympathies (especially if he goes ahead and pardons his sons and nephew). At the same time I don't see him opposed to many of the plans that Congress laid out in terms of Reconstruction in its initial moderate form, such as the Freedman's Bureau and the first Civil Rights Act (which would potentially butterfly the 14th Amendment as we know it).
Regardless of how it ultimately plays out, he isn't going to be seeking a term in his own right, and there is a good chance that he doesn't survive long past the inauguration anyhow given the increased stress he'd been under. There is a better chance that the National Union Party manages to hold together as an actual entity though far from guaranteed, and even if it should there are serious questions as to its longevity given the multitude of ideological conflicts within the "Tent".
If the National Union Party is still around, the question would be whether they are really competitive in an environment with both Radical Republican and Democratic candidates, and who Breckenridge's successor would be.