EDIT: According to @Ken Arnold in the other answer to this question, this technique will not work due to Time Machine backups being locked down by ACLs. I can't guarantee this, as I haven't checked it, but it's worth noting before you proceed with this answer. :ENDEDIT
Based on a quick look at my Time Machine backup, all you should need to do is take ownership of the files.
If you navigate to your Backups.backupdb, you should see a folder with the name of your machine on it. If you cd into that folder, you will find a long list of folders that correspond to every backup Time Machine has made.
Due to the way Time Machine works, each of these folders contains a complete representation of the state of your machine at that particular point in time. To get full permissions to read/write etc, you should simply take ownership of one of the folders.
Suppose you just want the latest backup folder, which OS X has helpfully created a symlink for. In that case, the sequence of commands (with a generic username on my machine) would be:
cd /Volumes/Backup/Backups.backupdb/Lukasa's\ Macbook\ Pro/
sudo chown -R USERNAME Latest
An alternative would be to simply take ownership of the entire backup history, in which case you would use:
sudo chown -R USERNAME Lukasa's\ Macbook\ Pro
An alternative would be to enable the root account and from there you can mess around with the files to your heart's content. Don't forget to disable root once you're done, and for the love of all that is holy don't ever type
I haven't tested this on my own Time Machine backup, as it's pretty extensive at this point and I'm quite attached to it. However, it's possible that some issues might occur with the nature of the Time Machine backup. In OS X, Time Machine uses hardlinks to reproduce unchanged files and directories. This means that each file contains the entire filesystem as it was at that point in time. However, OS X does not allow hardlinking to directories outside of Time Machine. I can't honestly say what affect this will have: possibly none, but I can't rule out unexpected behaviour. Just a warning.