To publish a video in Tumblr, follow these steps:
Create a custom field and make sure to make the field name: kapost_tumblr_video_embed
Create another custom field and make the field name: kapost_tumblr_video_caption
Add these two custom fields to the content type you are using to publish to Tumblr. See the article linked above for how to do this. Again, it should be your html content type, not your photo type.
When you create a new piece of Tumblr content, use the embed custom field to include the embed code for your video and use the video caption field to include the caption you want to use for your video.