Ok I'll break it down:
su - -c "/home/chris/t.sh" chris
insert that line into /etc/rc.local. That file is run as the last thing at startup by the system as root.
As root, you do not need another user's passwd to run another user's prog, assuming it has executable perms.
su - <-- means switch user
-c <-- means run the following string as a cmd
"/home/chris/t.sh" <-- cmd to run: use absolute path : string must be enclosed in quotes
chris <-- run as this user
Further: all the cmds in your script (t.sh here) must have absolute paths as well, inc paths to files read/written if any. All cmds/progs called must have exe perms.
The only reason you might need a passwd is if (one or more of) the progs you call require a passwd, in which case you'll have to (unfortunately) hard code them in and make your top-level script an 'expect' script:
Please add your distro to your profile. I'm assuming your distro uses /etc/rc.local. Some (eg RH) do, some don't, in which case they use a different but equiv file/method.