Best way to store data locally in .NET (C#)

I'd store the file as JSON. Since you're storing a dictionary which is just a name/value pair list then this is pretty much what json was designed for.
There a quite a few decent, free .NET json libraries - here's one but you can find a full list on the first link.

It really depends on what you're storing. If you're talking about structured data, then either XML or a very lightweight SQL RDBMS like SQLite or SQL Server Compact Edition will work well for you. The SQL solution becomes especially compelling if the data moves beyond a trivial size.

If you're storing large pieces of relatively unstructured data (binary objects like images, for example) then obviously neither a database nor XML solution are appropriate, but given your question I'm guessing it's more of the former than the latter.