While making the existing window larger is not a possibility, in most cases replacing it with a larger window is. It will require some adjustments to the wall frame to accommodate the larger window. Check the interior and exterior surface of the wall surrounding the window you propose to enlarge for obstructions such as masonry, drain pipes, or other immovable objects that may prevent you from enlarging the opening in the direction you desire. Adjusting your plan to make it as simple as possible is the best way to ensure success.
Pry the trim at either side and top and bottom of the window away from the wall with a pry bar. Remove the exterior trim in the same way. Cut the nails securing the window in the opening with a reciprocating saw. Cut through the exposed gap between the jamb and the frame along the bottom and up both sides, finishing by cutting across the top. Lean the window in and lift it down from the wall.
Mark out the new window size above, below or beside the existing window as required. Make the opening 5 ½ inches taller and 3 1/2-inches wider than the outside of the new window frame. Cut the drywall along this line with a utility knife and lift it out to expose the frame.
Cut any studs or other frame members that intersect with the outside edge of the new opening with a reciprocating saw. Remove the cut framing from the window. Cut and remove the drywall along the side lines of the new opening from the floor to the ceiling.
Cut two 2x4 studs to fit from the bottom plate of the wall to the bottom of the top plate with a circualr saw and position them in the wall set back with the inside face flush with the cut edge of the drywall at each side of the opening. Check with a level and adjust as needed. Nail on an angle through the top and bottom of each stud to secure to the frame with 16 d nails. These are your king studs.
Cut two studs 1 1/2-inches shorter than the bottom height of your window sill. Nail one against each side stud you just installed. Cut a piece of 2x4 to fit between the two king studs. Place it horizontally on top of the two short studs you just installed. Nail it in place. This is the frame sill.
Cut two studs 1 ½ inches shorter than the distance from the bottom of the wall's top plate to the top of your proposed window frame. Nail one on each side to the faces of the king studs with their top ends against the bottom face of the top plate.
Cut two pieces of 2x6 lumber the same length as the frame sill. Cut a piece of 1/2-inch plywood 5 1/2-inches wide by the same length as your frame sill. Glue and nail it sandwiched between the two 2x6s with all four edges flush to create a header.
Position the header with the wide face of the inside 2x6 flush with the inside edge of the king studs and its top edge against the bottom ends of the two short studs you nailed overhead. Nail on an angle through each end to anchor it to the king studs.
Cut two pieces of 2x4 to fit between sill and header and nail against the king studs on either side.
Set the new window in the opening, centred. Place a level on the sill of the window frame itself and add shims underneath to adjust it so that the bubble is centred in the indicator. Nail through the bottom sill of the frame into the wall frame sill at each corner. Wedge shims between window and wall frame at each top corner. Place the level on one side jamb, on the face of the window. Adjust the top of the window in or out as needed to level it. Nail through the window jamb into the wall frame in both top corners to secure it.