I like to use the tools that I carry on the bike to make sure I am not missing anything. It also ensures I've had some experience using them and learned their limitations.
1) Remove the left side cover by gently pulling on it
2) Use the 17mm socket to remove the bolt under the front of the seat.
3) Remove the seat by pulling it slightly forward while rotating up.
4) Use the allen wrench to remove the four bolts holding the seat mounting plate on.
I couldn't get enough torque for some of the bolts with the normal wrench. I put the end of the screwdriver on the wrench to extend it slightly and provide a more hand-friendly grip.
5) Remove the plate.
6) Use the 8mm wrench and the philips screw driver to loosen the bolt holding the battery tie down in place.
7) The top of the battery was blocked by a cable connected to this plug, so I found it necessary to disconnect it.
8) Use the philips screwdriver to disconnect the battery terminals negative (black) then positive (red).
9) The trick to removing the battery since there is hardly any space is to:
a) Hold the kick starter down with your foot to add an extra inch to work with
b) Shift the bottom of the battery forward and start to twist it
c) Get the bottom of the battery on the upper right side and the top on the lower left (see below)
d) Watch for snagging cables. It will just slide out with less than 1/8" all around.
e) It is a bit heavy, so don't crush your fingers.
10) Done! Reverse the steps to reinstall the battery.
This is a picture of my "original" Harley battery for reference. My original Russian factory battery was defective and replaced immediately after purchasing my Ural.
I would use the Odyssey PC680 Battery in the future.