The best way to stream imo is to have a 2 pctwo PC-setup. Streaming eats a lot of the cpuCPU power and there is no way you will have better stream quality with 1 pcone PC than with 2pca two PC setup. Unless you dontdon't have a pcPC with 2 cpustwo CPUs inside it, which is quite exotic and expensive as hell iI guess. And it's not for gaming.
So you can buy an avermedia live gamer portable lite (around 150$) to save some bucks and you dontdon't need any video splitter. Just make sure that you can use avermedia's hdmiHDMI input (check youyour laptop video output). And you will probably need an hdmiHDMI cable, if it is not goingdoesn't come with the capture card.
The question is - is your laptop capable of sending picture simultaneouslysimultaneously to the laptop screen and through the laptop's video output.
The difficulty about 2 pca two PC setup is sending audio to the second pcPC. You can resolve that thing with a 3.5 mm audio-cable splitters.
With the capture card, you will fit in 200$ I think, if you don't have a professional micMIC with xlrXLR or usbUSB output. Also there are some ways to send audio through the lan.