The President, through the Secretary of the Navy, really commanded the Navy. The concept of a general staff overseen by a serving officer did not come about for a few more decades. High-ranking officers, even admirals, were something like "the first among equals," respected due to their rank and accomplishments, but they had formal control only over their own commands.
Many historians have pointed to Gustavus V. Fox, the Assistant Secretary of the Navy, as the de facto CNO (Chief of Naval Operations), though this is of course projecting a modern role back onto a historical situation. It's unquestionably true that Welles depended significantly on Fox's experience as a serving naval officer, whereas his own experience was more along the lines of departmental administration (at which he was superb).
Technically speaking, the ranking admiral of the Navy was Charles Stewart, who had commanded the USS Constitution in the War of 1812, but I am not aware that it was anything more than a formality in his case. He held the real rank (on the Retired List) but I don't think he served in any real capacity during the Civil War.