Currently there are very few ways of increasing your clan rank. Clan rank goes up by accumulating clan XP.
You can get clan XP by adding people to your clan who have "Founder Status" - this means people who joined the for-pay version of Elite right around the Modern Warfare 3 launch. Since most of these people are in clans already, and it's a one-time bonus, this is not really a good source of clan XP.
The "real" way of getting clan XP is participating in Clan Operations:
Clan operations award clans that finish in the bronze tier or higher. Bronze requirements may change from op to op, but generally if your clan finishes in the upper half of an operation, it will be rewarded.
These rewards include badges that are displayed in your clan’s showcase and Clan XP.
Only the clan leader can enroll in Clan Ops, and in order to get badges or XP, you're going to have to have your clan members play the designated game type during the operations period. There's more detail on the "Compete" tab of Elite, under the heading "What are Clan Operations," which is just above the photos of upcoming Clan Ops.
The MW3 events team has had a spotty track record with scheduling and awarding Clan Ops to date, so if clan ranking/XP is important to you, you'll want to check in early and often for new ops.
Best Answer
Go to http://www.callofduty.com/elite and create a new account. Follow all the steps, then at the end link your Xbox Live or PSN account to your newly-created Elite account. Your old Elite account will no longer be associated. Log out and log in of Xbox Live or PSN on your console, and you're gold.