You need a "zipper" tool to unhook the lower hems of all the pieces. this is a small, cheap hand tool that will allow you to temporarily bend the lower edge back and unhook it from the piece below. To assist in reassembly, you should take a carpenter pencil and mark the top edge of each piece as it is revealed, during disassembly. The difficult part will be holding some of the pieces up and out of the way, to allow you room to work, without tearing huge amounts of siding off. You might try a helper, or scraps of 2x4 to wedge the siding up and clear of your work, Do some googling of how to renail it back up. Nails centered in the slots, and loose enough to fit a dime behind the head.
Be real clear that you understand the whole J-channel deal. It is a decorative trim, it never was, and never shall, be caulked in any way. It is there to give a place to hide the ends of the cut siding. Vinyl has tremendous amounts of dimensional instability, due to expansion and contraction when heated and cooled, so caulk will not stick to it for long, and many caulks don't stick to it at all. The goal is to remove enough siding and the Js to have access to the window flange. Next get a few rolls of 4" Zip tape. This is black tape that is sold by Huber lumber as part of their proprietary sheeting system, and there is nothing better for this application. Install the tape so it covers the housewrap and all but the inside 1/4" of the flange. The proper way is to do the bottom first, leaving the tape run past the window edge by 2" or so. Now do the sides and let them run long on the bottom, covering and sealing the bottom tape, and run them 2" past the top of the window. The top piece seals the top flange, the side tapes and runs past the side tapes a bit. This is called shingling. As water flows across all this tape, and down the wall, each layer covers the ends of the layer below. Good luck.
