The best way to stream is to have 2 pc-setup. Period. Streaming eats a lot of cpu power and there is no way you will have better stream quality with 1 pc than with 2pc setup. If you dont have pc with 2 cpu inside which is quite exotic and expensive as hell i guess. And it's not for gaming.

 So you can buy avermedia live gamer portable lite  (around 150$) to save some bucks and you dont need any splitter. Just make sure that you can use avermedia's hdmi input (check you laptop video output). 

Answer on top is ridiculous.    They say that lgp lite is too expensive (it's around 150$) and suggest you to buy a new pc, which is about 1000$. Wtf? Be smart , take the capture card and you will be golden. 

Good luck man!