SSH with a single, simple command argument will not start a shell, which means .bashrc does not get executed.
SSH with a compound command argument such as your second example should start a shell, but it will be a non-interactive one, so that won't run .bashrc by default. See Bash Startup Files in the Bash manual for more information.
However, calling 'source' in your second example command should work. Does your .bashrc have a line like
if [ "$PS1" ]; then or
if [ -z "$PS1" ]; then at the top? The sample bashrc does have one of these lines, and it prevents the script commands from running unless you are on an interactive shell. As mentioned above, SSH does not start an interactive shell by default when you pass a command argument. Try
ssh -t www.remotehost.com 'source ~/.bashrc; my_command', as this will force TTY allocation and act like an interactive shell.