I'm looking to use KeePass as my password manager, but I would like to have the nice cloud sync and browser integration that services like LastPass or 1Password provide.
I would suggest the same as the previous answer, by leaving the database file of Keypass on your Google drive and then utilize a NEW app called "CKP Keepass intergration for Chrome" in the Chrome web store. Nice app that will show the Keepass icon next to a website that you have listed in Keepass, then just click on the keepass icon and press "autofill" and it fills in user name and password. GREAT APP and very nice developer who is receptive to suggestions. I use it on my Chromebox and works great!
KeyPass stores its database of passwords in a single file. Solution: move the file to your Google Drive. Open it wherever you use KeyPass.
A plugin called ChromeIPass integrates KeyPass directly into Chrome. KeePass also offers an auto-complete feature built-in.
There are even options to access your KeePass passwords on your phone.