How to add /home/username/bin to $PATH?

To do that you need to type in your terminal:

export PATH="$HOME/bin:$PATH"

This change is only temporary (it works only in the current session of the shell). To make it permanent, add the line to your .bashrc file located in your home directory.

Ubuntu (and Debian based distros) automatically add $HOME/bin to the PATH if that directory is present. You can check this in ~/.profile:

# set PATH so it includes user's private bin if it exists
if [ -d "$HOME/bin" ] ; then
    PATH="$HOME/bin:$PATH"
fi