iPhone 5c says its out of storage
It sounds like you're looking at your available iCloud Storage rather than your device storage (which is different than 5 GB). You may have, for example, 15 GB out of 16 GB used on your phone (depending on total storage).

If you are trying to update many apps at once, or a very large app (there are some in the gigabytes), you need enough storage on your phone, and deleting a few small apps may not be enough. Try deleting apps that use more space and try updating just one or two individual apps that are small (in the Updates page of App Store it says the file size for the app updates) and see if you're still getting an error.