These previews in Skype and similar programs are generated based on the meta tags.
Different applications/social networks use different meta tags for this. Skype, for example, understands the open graph meta tags: http://ogp.me/
There are many articles describing the process of writing meta tags. You can take a look at this two:
https://warfareplugins.com/open-graph-tags-twitter-cards-rich-pins/
https://moz.com/blog/meta-data-templates-123
What you would need is a renderer, that takes the link, parses the page, tries to understand the meta tags and renders the preview based on the meta tags.
Some of the open source implementations for open graph are listed here: http://ogp.me/#implementations