Or you can select all your contacts in Address Book, click File/Export/Export vCard. Save the file to your desktop. Open Contacts at iCloud.com then click the Gear icon in the lower left corner to reveal the Import vCard option.
It will get your contacts to iCloud and syncing with an iOS device however obviously any changes will not automatically sync between iCloud and the computer.
I'm doing what you said now (since I too have 10.6.8). Is it supposed to be a long process? I have only 43 contacts in my Address Book, and it turned them into a vCards file in a second or so. But Contacts at iCloud.com has been spinning its wheels for the last 10 minutes and appears to be happy to continue.