Download 6.2.3, and follow the instructions in UPGRADING.html for VERSION 6.2.1 to 6.2.2. The instructions haven't been updated for 6.2.3, but the only files changed between 6.2.2 and 6.2.3 are:
config.inc.php
install.php
ubbt.inc.php
admin/editconfig.php
Since you're upgrading, you won't be using install.php, and you don't want to overwrite config.inc.php.
The instructions refer to CHANGEDFILES.txt, but I never trust that. I either upload all the files, except for the ones that contain customized settings, or I use BeyondCompare to see what's been changed.
Of course if you have hacks, then you have to deal with merging your hacks into the changed files, or vice versa.