The illustrations used in the stamps and Vines were created by Grahame Baker-Smith. The Vines, which were created by PR agency Onlinefire, highlight key scenes from the story.
For the first time, Royal Mail has also teamed up with children’s publisher Walker Books to create a pop-up book of the story using the illustrations.
This is the second time Royal Mail has used Vine in a bid to make stamps more appealing to a younger audience. Before Christmas, it used Vine to promote its 2014 Christmas stamp collection.
Head of marketing at Royal Mail, Fraser Chisholm, said: "Alice’s time spent in Wonderland is a story the world knows well. We wanted to refresh the magic of Wonderland by making it accessible via communications technology enjoyed by many people today.
"Video is a great way to capture imagination and Vine allows you to do it in small chunks that make people want to watch again and again.
"Stamps are not just for collectors. They also appeal to people interested in the subjects featured on our special stamps."