How can I make a backup of my operating system to reinstall on a replacement hard drive?

I have a 2009 aluminum unibody MacBook Pro running OS 10.6.8 Snow Leopard. The hard drive is failing and I have not been able to do a Time Machine backup in about 6 weeks. I have individually copied all my important data files and archived my calendar, contacts and email mailboxes to external drives. Now, I want to install a new hard drive, restore my last successful backup, then supplement it with more recent data from my external drives.

Once the new drive is installed, I presumably need the operating system to accomplish the rest. How can I copy the current OS, along with all necessary files, to an external drive, then reinstall it onto the new drive? Although I purchased my MacBook Pro new from an Apple store, it did NOT come with CDs or DVDs of the OS.

Thanks in advance.

Everett