I have a SharePoint 2010 site that has Kerberos auth properly configured and is added to my "Trusted Sites" in Internet Options.
Using KerbTray I get a Kerberos ticket on both Chrome and FireFox. I purged tickets before each attempt.
I then opened SQL Server Management Studio and connected to the SharePoint DB server. I opened a new query window and used the following SQL:
FROM sys.dm_exec_connections c
INNER JOIN sys.dm_exec_sessions s on s.session_id = c.session_id
I opened FireFox and watched my session authenticate and saw my domain user account hitting the system with Kerberos. I did have to configure FireFox to use Windows Auth, but I am pretty sure recent versions of Chrome automatically use Windows auth and will understand Kerberos just fine. Is your site in "Trusted Sites"? That's all I can think to have you check.
EDIT: I just saw this blog post
In DNS are you using a CNAME or an A record for your SharePoint site's domain name?
You can check using
nslookup via a command prompt:
Alias sections. A CNAME is an alias. You want an A record (Address record).