The only information you can legally request removed from your credit report is information that is incorrect.
If your credit report correctly reports that you had a debt and that it was behind and sent to collections then you can't legally challenge or change anything. They report the truth and this is a case where they are right and doing their job correctly.
If they continue to report that you owe them money (Balance is reported as $10,000 in January 2014) but you've paid them off in November 2013 or they report that you only paid off half or some other change that is incorrect, you could dispute the account because it is now in error. This is done by filing a form with the credit reporting company that has the error and they have 30 days to remove the information or confirm that it is correct and provide you a way to appeal. You'll have to do this with each credit reporting company that has an error.
Anecdotes exist that show the credit bureau will remove the entire account when correcting an error and anecdotes exist that the credit bureau will fix the error, leaving the account. If this is your scenario, I don't know which solution you'll end up with.