We've gate checked our DD's wheelchair many times, so I know a lot about gate checking. The posters who said you are not likely to get your stroller put into the closet are right. Some of the closets are too small and the first class passengers expect to be able to put their coats and hanging bags in there, so a stroller going in there would be very unusual. We have gotten DD's wheelchair in the closet a few times (they are required by law to store wheelchairs in the closet if asked), but we've had to do a lot of "discussing" with the flight attendant to get it in there.
It's best to ask about a gate check tag for your stroller when you first check in for your flight. They may give you one right away or tell you to get one at the gate. Don't wait until you are loading; ask for one as soon as you see a gate agent. The tag needs to be marked correctly or your stroller won't be routed to the right place. If you wait until boarding, there is more chance of error.
You can wheel the stroller right to the gate; you are responsible to fold it and secure it so it won't unfold and then the baggage agents will load it with the baggage (the very last things to go in). Make sure anything that might open, like a canopy, is secured. If it unfolds, it might bend. Some people use bags of sleeves with holes on the bottom and top to help protect the stroller.
The last part is remember to pick up your stroller when you leave. We are often the very last people off the plane and we almost always see 1 or 2 strollers, left there by people who forgot they gate checked them.