An external optical drive will work fine, although it will drain the battery faster if you're not plugged into a charger. But you have to be careful about which external drive you buy.
Do not buy Apple's USB SuperDrive; it will not work with your computer!
I learned this the hard way last week. Apple has apparently crippled the firmware for the USB-SATA converter in the drive enclosure so that the drive will only work with Macs that came without optical drives (MacBook Airs, MacBook Pro Retinas, late 2009> Mac minis, late 2012> iMacs, current-gen Mac Pros). It's not just that you can't boot other computers from a system disk in the drive; you can't mount disks at all. Booo, Apple!
Instead of spending $80 on an Apple external drive that doesn't work, spending $40-50 less on a third-party drive that does work is fine. In my experience, iTunes and DVD Player auto-launch on most external drives, regardless of who made the drive or the enclosure. You might even remove your failing internal drive using iFixit's guide and try it in an external enclosure, to see if it works that way. If you buy a drive and an enclosure together, that gives you the opportunity to replace the internal drive as well.
You can get an external Blu-Ray drive, which will also play/record CDs and DVDs. Apple does not directly support Blu-Ray disks, so DVD Player won't play Blu-Ray movies. There are several commercial third-party Mac Blu-Ray players available; the only free one is the OpenSource VLC Player, which plays a wide variety of media formats, including Blu-Ray.