Holmes's client in "A Scandal in Bohemia" is the King of Bohemia. His goal, in this story, is to secure a compromising photo of himself with Irene Adler, even as she currently is threatening to release the photograph (as he himself is about to be married). He turns to Sherlock Holmes for assistance in retrieving the evidence of his affair.

Ultimately, his entire purpose in coming to Holmes is to avert a scandal (one that would sabotage his marriage). This is actually why he arrives wearing a mask while also adopting a pseudonym. If the King of Bohemia were to arrive in London to call upon Sherlock Holmes, that would cause a great deal of speculation and public interest as to the reason of his arrival (speculation which he would very much prefer to avoid). On the other hand, as he himself states to Holmes, the knowledge of his affair with Irene Adler is too sensitive for him to entrust with an intermediary. Therefore, he comes himself, pretending to be his own agent, wearing a mask in an attempt to safeguard his identity.

His goal is to contain the information of his affair, and with this in mind, he has an interest in avoiding as much public attention as he possibly can. This is why he turns to secrecy and disguise.