Cut the caulk line around both sides of the casing and the window sill itself. Drag a razor knife top to bottom to do this, making sure to get the full caulk line if you can.
Use a crowbar to pull out the casing from the drywall. If prying, make sure to use a brace against the wall so you don't dent the drywall.
Use a prybar or chisel to slowly pry the sill up. A hammer also helps. Try to keep the sill in one piece so you can use it as a model when cutting the new sill. Measure the sill area and use the old sill as a template if possible. Cut the new sill to size. Dry fit several times as the walls is likely not be completely straight.
Use nails or brad nails to install the wood board into place ensuring that it is nailed into the framing below.
Reattach or replace the side casing and trim that was removed with nails or brad nails.
Complete the finishing of the window. Fill and smooth all nail holes with wood filler and patch walls if necessary. Caulk around the edges of all the window trim and frame. Prime and paint as needed to match the existing trim in the house.