I had a similar problem with a canon powershot camera which turned out to be a faulty memory card.

Does the flash fire when off the camera? Are the batteries fresh? I have found that most not firing problems are due to dirty contacts on either the flash or the camera, however be careful when using electrical contact cleaners as it can melt plastic. Spray on a cotton bud and carefully wipe the contacts. I hope this helps.

The 5d doesn't have a built in flash. You need to fit an external flash!