I'd use JSON.NET.
http://james.newtonking.com/json
MIT License which means if you're doing anything commercial you are good.
I don't think you would want to go straight to a Dictionary, because there is some stuff there that isn't a one to one relationship. So you could make a class like the following.
public class BitlyData
{
public string LongUrl{ get; set; }
public string Url { get; set; }
public string Hash { get; set; }
public string GlobalHash { get; set; }
public string NewHash { get; set; }
}
You could then use Json.NET to turn that String into an JObject. So we'll call your string bitlyString.
JObject bitlyObject = JObject.Parse(bitlyString);
Now we have that all that is left to do is access the data.
BitlyData theData = JsonConvert.DeserializeObject<BitlyData>(bitlyObject["data"]);
Then you can access the url (and any other pieces) using the getters.
Of course you could make it even better by having class that handles the other bits as well so you just do one serialisation.