Had the same problem recently (on an iPhone 4S, iOS 5.1.1). The phone couldn't back up to iCloud as the storage limit had been exceeded, and tapping "Manage Storage" in the iCloud settings didn't work: it just briefly showed a spinning wheel but I couldn't enter the Manage Storage screen. Looks like a bug.
In the end I was able to solve it via the iCloud settings on a laptop, as Winston Churchill suggested. Using this route, I removed the backup that was clogging the available 5GB free storage, after which I was able to access Manage Storage again to tweak the settings. In my case this meant disabling iCloud backups of photos in the camera roll, which was the reason for the large size of the backup file in the first place.
Here's what I did:
1. Important step: first of all, make a backup of your phone in iTunes by connecting it to the computer and chosing the backup option in iTunes. You will delete the iCloud backup in step 3, so it's best to have another backup handy. It's not very likely your phone will crash in the short time it takes to complete these steps, but better safe than sorry.
2. After making the full phone backup, I downloaded all the photos from the camera roll onto my computer and deleted them from the phone. There are different ways of doing this, on a Mac you can for instance use Image Capture. This step is not really necessary to get the backups working again as you will probably disable camera roll backups in step 3 anyway, but it's good to get into the habit of backing up photos on the computer regularly, rather than in iCloud. If you want to keep some photos on the phone as well, you can always copy them back from the computer onto the phone.
3. Access iCloud storage management via the computer rather than the iPhone. Instructions: http://support.apple.com/kb/ht4847 > Backup > Deleting backups and turning off Backup for your device > Mac OS X (or Windows 8, Windows 7 and Vista). If you can't or don't want to upgrade your MacOS version, you can access your iCloud storage management from any other computer with the right version of MacOS or Windows, by logging in with your AppleID. If you use a Windows computer and the iCloud control panel hasn't been installed yet, you can find it here http://www.apple.com/icloud/setup/pc.html.
I don't think you can delete specific elements like the camera roll from the backup this way, you have to delete the whole thing. But you made another backup in step 1 so this isn't really a problem.
4. On the iPhone, try to access Settings > iCloud > Storage & Backup > Manage Storage again - it should work now. In the next screen, choose your iPhone and you will get the Backup Options where you can switch backups on and off for different apps. It shows you how much storage each app takes, you will probably see that the Camera Roll is the largest so you may want to switch that off. Once you're happy with the settings, go back two screens to Storage & Backup and switch Icloud Backup back on (it was automatically switched off in step 3), and if you are on Wi-Fi, tap Back Up Now so it makes a new iCloud backup right away.
That's all, hope it solves your problem.