They can typically run about three riser joints per hour. Each joint is either 75-ft (or 90-ft depending on riser design) in length. They are connected by a bolted flange typically. However there are other designs that can be used to increase the running rate.
Essentially they just connect new joints on top of each other until they reach the bottom. At that point they use a special joint called a landing joint to connect the stack to the wellhead. Then they unlock another special joint called a telescopic joint (TJ) to allow compensation of the vessel motions. The outer barrel of the TJ is connected to a tensioning system (either direct acting or wire-line type) to provide the necessary top tension to keep the riser stable.
There are of course more steps involved but that's the short version.