I'm answering to both your original questions and to your comment "There are an awful lot of admirals with virtually no backstory. Are there any with any?"; so I'm providing also background info about What they did before appearing on-screen? along with What they did before becoming admirals?
Admiral Leyton was previously Captain of the USS Okinawa during the Federation-Tzenkethi War, and had Benjamin Sisko as Executive Officer. They generally had a good working relationship, and Leyton appreciated Sisko's honesty, despite being a decisive commander.
His previous relationship with Sisko led him to try to enlist the commander of Deep Space 9 in his attempted coup d'état against the Federation's President.
Admiral William Ross was the Starfleet military leader during the Dominion War. He commanded from Starbase 375.
DS9 scripts described him as a man no older than Sisko, [that] shows signs of the burden of office. Barry Jenner, the actor that portrayed him, said that Ross "was obviously a line officer who'd seen a lot of combat, seen a lot of people killed, and he's got a lot of memories".
It seem that Ross had a career as a military leader, a field where he acquired great experience at a great personal expense; that possibly had a role in his promotion to Admiral, and to him being chosen for the role of Starfleet Commander during the War.
Admiral Owen Paris was previously Captain of the USS Al-Batani, and took part in the mission known as the Arias Expedition.
He held several high ranking posts in Starfleet from captain to admiral, and was notably known as a Starfleet Academy instructor, where he also taught to his own son, Tom Paris.
Admiral Mark Jameson, while holding the rank of Commander, was sento to negotiate the release of hostages on Mordan IV. His mission was successful but controversial, as he violated the Prime Directive and many Federation principles.
He advanced through the ranks until he reached Rear Admiral, his last command was the USS Gettysburg. He contracted Iverson's Disease and was eventually confined to wheelchair.
Admiral Erik Pressman was a previous Captain and commanded USS Pegasus (William Riker was serving here too, as an Ensign), a ship used as a testbed for various technologies, most notably a phasing cloaking device. These experiments were illegal, and he faced an inquiry, but had no further consequences.
He obtained a position within Starfleet Intelligence, and later was given the rank of Rear Admiral.