Assuming you are using a fairly recent TurnKey appliance release; we preinstall drush (DRUpal SHell - a commandline drupal tool) in both of our Drupal appliances (Drupal7 & Drupal 8). Actually we use drush to install drupal. So the quickest easiest way to update drupal would also be via drush.

TBH, I'm not particularly familiar with drush myself, but reading from the Drupal 7 docs, in conjunction with the brief documentation we've provided; it should be pretty easy. So for Drupal 7 something like this should do the trick:

#create backup first and test it!
drush vset --exact maintenance_mode 1
drush cache-clear all
drush pm-refresh
drush pm-update --simulate #simulate first to check for issues
drush pm-update #if you just want sec updates; use '--security-only'
drush vset --exact maintenance_mode 0
drush cache-clear all

FWIW, personally the way I would do it, would be to create a backup of your current production server using TKLBAM. Then download the relevant appliance ISO or OVA and install to a VM (e.g. VirtualBox). Then restore your backup to your local VM (testing your backup in the process). Then do a trial run of the update in your VM. That will give you a clear indication whether it will "just work" or whether you'll need to do further investigation.

Please note, that if you are using a TurnKey Drupal appliance that is NOT v14.x (e.g. v13.0), then you will likely hit obstacles. Now is probably as good a time as any to migrate to v14.2. Note that you will almost certainly need to manually tweak some settings when you restore your backup to the local v14.2 VM. We have a doc page that covers a suggested workflow when migrating appliance data to v14.x. When you have that working as you want, then migrate to a fresh v14.2 Drupal appliance in the cloud (assuming that's where your current server resides).

If you are using Drupal 8, then I would give extra encouragement to follow my above advice. AFAIK Drupal 8 should work using the same commands noted above, but I can't 100% guarantee (hence why the additional encouragement to test it first). Again there are Drupal docs; but the Drupal 8 ones aren't so user friendly.\

If you are using a really old Drupal 6 appliance; then there isn't an easy migration pathway. My guess is that you'll want to find yourself a Drupal expert to help you with that. Again though, I would recommend that you take the opportunity to also migrate to the current release.