Put `.vimrc` into the `.vim` folder

From the Vim man page:

-u {vimrc}

Use the commands in the file {vimrc} for initializations. All the other
initializations are skipped. Use this to edit a special kind of files.
It can also be used to skip all initializations by giving the name "NONE".
See ":help initialization" within vim for more details.

If you still want your ~/.vimrc to be processed, try this:

vim -u ~/.vimrc

Add the following line to ~/.bashrc (or your shell's equivalent file if not bash) to have the -u switch added automatically:

alias vim="vim -u ~/.vimrc"

You won't be able to add something to ~/.vimrc to prevent /etc/vimrc from being read, because the system file is processed before your user file (see ":help init", section 3, "Execute Ex commands, from environment variables and/or files").

You don't. Symlink it instead.

mv ~/.vimrc ~/.vim
ln -s  ~/.vim/.vimrc ~/.vim

You cam do the symlink thing to sync your vimrc through Dropbox as well.