@nashtrik :-
Couple of options, really. Either Ozsouth's 'traditional'-type package build, which Bigpup linked to. Or, there's my Google_Chrome-portable build.....completely self-contained, running as user "spot" and, as the name implies, you can move it around to any location you want. You can even run it from a flash drive!
viewtopic.php?f=90&t=146
If you decide to give this a try, it's always a good idea to save your bookmarks to an HTML file before switching. The 'portable' doesn't use the 'standard' location for its profile, so it won't pick up & use your existing one.....on top of which, since your current version is nearly 20 'main' releases behind, the browser will bitch about using your existing profile anyway.
That request for your user password is because the .deb package, as is, expects to find itself on a 'mainstream', multi-user system.....complete with clearly-defined (and password-protected) 'user' directories under /home. Which is NOT the standard Puppy setup.
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However; please remember one thing for future reference. The "HOW-TO" section is NOT for asking "How do I do such & such?" It's for members who wish to share something they've discovered, or a different way of doing something.....and want to let the rest of the community know about it. Okay?
EDIT:- It's not a good idea to install Chrome from the deb packages. It has a nasty habit of breaking certain permissions in Puppy.....most noticeable is sudden loss of the ability to print anything. And from late-60s Chrome onwards, the browser refuses to run as /root; if it cannot find a user directory under the /home directory, it won't even install. Puppies, generally speaking, don't use the /home directory, so it isn't present.
More recent Puppy packages of Chrome - in various different formats - are built to take this into account. Those of us who build the various Chromium-based browser packages DO go to a lot of trouble to ensure fuss-free operation for community members, believe it or not!
Mike.