The CardDAV sync method recommended by KiltedTim works, but over time it can lead to more problems than it solves (at least it did for me).
For one thing, some of the iPhone contact fields are not supported. It looks like everything is fine backed up on your Google account, but when you setup another device or use Google contacts via the web you find all kinds of things missing... Later if you upgrade your iPhone you'll have even more problems. For example, Siri does not like working with the sync'd Google CardDAV contact accounts at all. If you say "call my sister" (use a relationship not the name), Siri will not be able to remember who your sister is and ask "who is your sister". For some reason, Siri will only remember who you are and who your wife is. All the other relationship labels won't stick for long. Siri seems to keep the relationship information in some kind of short term memory, but it eventually evaporates.
Unless you absolutely must have instantly updated and sync'd contacts I suggest you use iTunes to sync your contacts to Google instead. This way you will be able to use all the features your iPhone has to offer and all the sync-able data will be sent to your Google contacts (either way you have to move the non-sync-able info manually). If you enable wi-fi sync on your iPhone this will probably happen at least once a day (when you get home) with 0 effort from you.
To use iTunes to sync your contacts plug your iPhone into the computer your phone is sync'd with and select it from the devices shown in iTunes. Go to the "Info" tab for your iPhone, check the "Sync Contacts with" box and change the drop-down from Outlook to Google.
It's far from perfect, but this is the best solution I've been able to get. I'm using this setup to sync one Google contacts account across multiple iOS devices. Though if anyone out there knows the way to get the both instantly updated Google contacts and 100% Apple/Siri compatible contacts please speak up!