Place shims in the rough opening along the bottom sill. There should be a 1/4-inch gap to allow for expansion and contraction. Use shims in each corner and about every 1 foot to properly support the window.
Lift the replacement window into the rough opening and tilt into place. Ensure that the sill is evenly supported by the shims.
Use shims on all four sides, to get the window roughly square in the opening. Trim with the utility knife as needed, and apply a drop of caulk to each shim before placing it. Make sure you place shims at each pre-drilled pilot hole along the sides.
Check the window's placement using the bubble level, both horizontally and vertically. Tap the shims with the hammer to make minor adjustments, if needed.
Apply a drop of caulk to a screw, and drive it through one pilot hole in a lower corner to secure the replacement window to the jamb. Check the window with the bubble level one more time to ensure it's square, tapping shims with the hammer to make minor adjustments.
Fasten the window at each corner in a similar manner, then fasten through any additional pilot holes.
Fill air gaps from the interior, on all four sides, with low-expanding spray foam. Larger gaps may be filled with fiberglass insulation. Cover each screw head with caulk. On the exterior, caulk any gaps with acrylic caulk.
Replace the interior window stop and any siding you may have removed.