Unfortunately what you get is download, followed by local playback.
However there are a few workarounds that can help get you closer to where you want to be.
Option 1: YouTube
If you first upload your video content to YouTube, then share the resulting URL as a Link in Chatter, Chatter embeds the video within the Chatter feed. This gives the impression that the video is actually in Chatter.
The downside of this is that YouTube videos are public, more often than not, and in some cases your company might be blocking YouTube entirely.
While YouTube allows some privacy, it can be somewhat inflexible. Currently you have 2 options "unlisted" which means that a user needs to specifically know a given video URL. This would be practical for non-sensitive videos, however most would consider this to be "security-by-obscurity" (aka not secure). The other option is "private", this would have promise, however anyone you share your video with must already be a YouTube user. Not very practical for corporations.
Option 2: Vimeo.com
Vimeo offers an interesting take on video sharing, unlike YouTube, you can add a simple password to a video to limit the viewing audience. While not as secure as having the video behind your firewall, this does offer a slightly better security experience.
Step 1: Upload your video to Vimeo and click the Video Settings button (gear icon in the top left). Several options exist, including the option to password protect.
Step 2: Grab the link using the Share button in Vimeo
Step 2: Create your Chatter post, use the link that you created in Step 2 as the Link in your Chatter post
Notice that the video thumbnail and event the video name are obfuscated.
Step 3: Click play on the video and the user is prompted for a password
After the password is provided, the video plays inside the Chatter feed
While Salesforce.com does not (yet) provide native video embedding in Chatter, such functionality can be achieved by leveraging the services of YouTube or Vimeo. Vimeo has the advantage of accommodating individual video paswords.
Salesforce.com leverages the services of Embed.ly to provide the rich media linking and embedding. For a comprehensive list of all supported providers, see http://embed.ly/embed/features/providers