You should have UOAssist running at this point. If you don't, start it up.
Now log all the way into the game with any of your characters on any of the official OSI
servers. If you aren't in a safe place you can log right back out as UOAssist will
save the seed it made during the logon procedure. Press the '?' button in the bottom
right corner of UOAssist (circled in red below). If you are running UO in full
screen mode you'll need to either switch to windowed mode (alt-enter) or hit alt-tab to
see the UOAssist screen.

After press the '?' button you should see the About screen. Press the button
labeled "Registration ..." on the About screen. This should bring up the
following screen:

The seed is found in the box right near the top. In the screen shot above the box
is blank and has a gray background. On your screen you should see something like
^7ac~,tUaq,Yu or something funny looking like that. If the seed box says '<You
must login to Ultima to obtain a seed>' then log out of the game and then back in
again. Seeds are very 'sensitive' so it is important that you use copy and paste to
move the seed from UOAssist's registration dialog to the spot on the trial registration
form in your web browser. Again it is VERY important to use the exact seed, using
the copy (select the seed then press ctrl-c) and paste (ctrl-v) method ensures accuracy
(copy is ctrl-c and paste is ctrl-v in most Windows applications, if ctrl-v doesn't work
in your browser see if you can find Paste on the Edit menu and use that).
You might note the blue box near the bottom of the above screen shot. That is
where you'll need to put the key once you receive it. Don't forget:)
A few extra notes on what the seed is or isn't. The seed does not contain your
account name and/or account password. It contains an identifier received from the
servers so I can make unique seed/key pairs. The information in the seed does not
allow access to your account or access to your private information from OSI's registration
site. You can change your account name or password at any time and it won't affect
your seed or key. Also if you have multiple machines you will notice that seeds for
the same game account are different. This won't affect your key at all and is
normal. If you have multiple game accounts please read
this.
Press your browsers back button to return to the steps.