Fine soldering experience is required. This isn't a soldering tutorial. Work carefully, assume everything inside is fragile. You should have received a board, 3 flex cables, and a USB cable. You'll need a small Phillips screwdriver (#000), soldering iron with a fine point (and solder). For case cutting you'll probably want a Dremel tool with small cutting bit, X-acto knife, etc. Tweezers will also be useful.
Remove the two rubber feet. Unscrew and remove the back cover. ⚠ Two cables for the shoulder buttons need to be disconnected, be careful not to tear them off while opening.
Remove screws, and disconnect cables from the mainboard. Small black connectors: Pull cable out horizontally. Small connector with gray latch: Lift latch to vertical position, then slide cable out (& close it to protect it from breaking off.) Wide connectors: Flip tab up, then slide cable out. Pull out the analog stick and disconnect its cable.
Flip open the top screen, place the 3DS face down. Flip the mainboard over (carefully, cables in the top corner are still connected). Remove the cartridge slot. Remove the plastic piece covering the bottom screen. Cut off the top right corner (about 2cm x 1cm). Put the plastic piece back into place (watch the Home button, it falls out). Put the cartridge slot back on and screw it down.
Disconnect the top corner cable from the mainboard (leave the others connected). Open the black latch on the (U shaped?) flex cable. Connect it face down onto the mainboard. Fold it over, insert the disconnected cable, gold side facing down. Close the black latch, locking the cable into place. Connect the L-shaped flex cable to the bottom screen in the same way, as pictured.
Flip the mainboard over. Reconnect all cables. Put screws back in (all except one, where the capture board will go.) Soldering: Tin the 6 points (pictured) with solder. Position the small L shaped flex cable over it, apply soldering iron to tack it into place. Solder the round pad down to give extra mechanical support. Now is a good time to check that everything still works. Hold the battery in place and turn the 3DS on, check that both screens and audio are still working.
Use a dremel, etc. to carve out space for the capture board.
{ .. attach capture board, flip over, screw it down, attach other small cable. connect analog stick, screw it down. }
Solder wires as shown for USB charging.
{ .. connect shoulder button cables. If possible, tuck cable under the capture board (keeps it from getting pinched between board & case). screw case down. adjust screw near capture board as necessary so the USB connector is level. put rubber feet back on -- one side will stick out more, slice it so it's level with the other one, use a bit of glue to keep it in place }