Based on evidence from the book, the reader can only assume what Chillingworth sees when he opens the sleeping ministers shirt; however, based upon his Dimmesdales extreme guilt and unusual actions conclusions can be drawn that lead us to believe that the distraught minister has actually branded himself with a letter "A".  This conclusion is later proven, as people actually see the brand in his flesh.

The book only says that he sees a mark on his breast which confirms his suspicions, and leaves it up to the reader to imagine what the mark could have been at that part in the book.

Later, in the conclusion, it says:

Most of the spectators testified to having seen, on the
breast of the unhappy minister, a SCARLET LETTER--the very semblance of that worn by Hester Prynne--imprinted in the flesh.

How it got there is left up to the reader's imagination, but several possible scenarios are given. Check out the link below to read that part: