Let me preface this by saying I've never used TFS, but the functionality that you're describing seems logical to me based on what a Definition of Done is.
A Definition of Done is something that applies to Product Backlog Items as well as Increments. Each can have its own Definition of Done - what it means for a Product Backlog Item to be "done" is different than what it means for the Increment to be "done". In both cases, it's simply a convention - it's not part of the backlog item itself.
In my experiences, the team's current Definition of Done is captured somewhere visible, like a wiki page. Over time, the Definition of Done for a team (or one that is shared across teams, if there are multiple teams working on the same product) will evolve as the team(s) mature.
Since the Definition of Done is general (a single Definition of Done for all Product Backlog Items and/or a single Definition of Done for the Increment), you may choose to add specific considerations on a particular Product Backlog Item to ensure that you meet the Definition of Done if there is some non-obvious work. Likewise, if there is particular work that needs to be done to ensure the Increment meets the Definition of Done, tasks can be added to the Sprint Backlog. But in most cases, I would find this redundant and the team should strive toward simply meeting the Definition of Done in their daily work.