For streaming I would advise 2 things:
First: Get some cheap headphones, the Sennheiser 201 cost 15 to 20 bucks and are better than many headphones for 100 bucks.
Get a Mixtrack Pro or a Beatmix 2, yes it has a soundcard.
Now there are 2 possibilities:
Either get a copy of Virtual DJ Pro (quite costly) as it can do streaming build in. All you need to do is configure the broadcast server within the software and it will stream to that server AND use the soundcard of your controller simultaneously.
Or you use another DJ software (Serato Intro comes with the Mixtrack Pro and Beatmix 2) and use an addon sound capturing and broadcasting like Simplecast (PC) or Nicecast (Mac).