Turns out you have to edit the key bindings for the
emacs-copy table. Here's how I made the changes that I wanted above.
First, I checked out what the current bindings were by entering the following command in tmux (by pressing Prefix+colon)
list-keys -t emacs-copy. This gave me a list of the keys that were bound in emacs-copy mode and I looked through them to figure out which commands did what I wanted.
Then, I added the appropriate key bindings to my ~/.tmux.conf file like so:
bind-key -t emacs-copy M-j cursor-left
bind-key -t emacs-copy M-i cursor-up
bind-key -t emacs-copy M-l cursor-right
bind-key -t emacs-copy M-k cursor-down
bind-key -t emacs-copy M-u previous-word
bind-key -t emacs-copy M-o next-word
Finally, I ran the tmux command
source ~/.tmux.conf inside tmux to get the keys bound without having to exit and reload the session.