All you need to do is create a skill, which is much easier than it sounds -- the whole process took me less than 5 minutes.
Setting up your Alexa greeting card
To create your greeting, go to Skill Blueprints and click on the Hallmark Holiday Greeting option. Follow the onscreen directions to pick a theme and record your message. Then, click Create Skill.
Sharing your greeting
Once Amazon processes your skill (it can take a minute or two), click on the Share With Others option on the right side of the screen.
Here you can choose how you want to share your holiday card, including over email, with Facebook or using a link you can copy and paste anywhere, including a text message.
Now, the recipient of your card needs to enable the skill for their Alexa device. They can do that by clicking on the option provided with your message that you just sent with the Share With Others Option.
Once the skill is set up on their end, they can say, "Alexa, open Holiday Greeting" to hear (and see if they have an Alexa device with a screen) your holiday message.