To give another deviant a direct link to this FAQ answer, use the following code in your comment or forum post:
Can I delete my deviantART account?
Accounts cannot be deleted! We don't ever want you to leave, we'd miss you too much, sniff.
However, should you feel that you really really really REALLY must go, or you need to remove some personal information, please follow the instructions below.
-----------
In order to remove information and submissions from from your account please see the following instructions:
1.) Journals: To delete your journals, you will need to visit your journal page, once there you will need to click on 'Edit Journal Entries'.
3.) Storing Deviations: If you elect to store your deviations rather than delete them please visit your Gallery management page.
4.) devWATCH (Friends): To remove users from your deviantWATCH, you will need to visit your devwatch management page, from there you will need to check the checkbox titled 'remove' next to each user, once done click on 'Update'.
5.) Favorites: To remove favorites, you will need to visit the deviation you have favourited and click on the 'Remove Favorite' link, located to the left of the deviation.
6.) Personal Details: To remove your email and others personal details you will need to visit your Edit Settings page. To remove your full name from your userpage you will need to visit your userpage and hover over the left section of your stats area, once clicked the text will change to about me, clicking this will bring up any options which you may edit.
7.) Avatar: To remove your avatar please save the default image and replace your current avatar in your Profile page on the Identity tab.