You first need to setup forms with the right permissions - edit.own
needs to be assigned to the group that your users will be a part of.
Unfortunatelly there is no delete.own etc in Joomla, so you can only
allow or deny delete/create on all items.
Then you can make a
list of articles filtered by the Article author field (and set to live
value user->id), so that each user will only be able to see his own
articles. On each list item add edit button and other buttons as
described in the manual (manual is written for admins than see all
items, you need to apply additional filtering as described)