1.Download the free software OBS. In the page you'll find all the steps necessary to configure it according to your PC's "strength".
2.Once you've configured OBS, add as video capture source your webcam or camera. you can even use a laptop webcam. Then also add a window capture and set it to record LiveSplit or whatever program you use.
3.Edit your scene so it looks nicer. Cropping the splits and the video capture so it looks smooth and nice.
4.Stream. you'll have to do some testing with your streams until you get the perfect bitrate to avoid lag and others.
If you need more help, OBS has good guides better explained than this. If you tell me your PC settings (graphics card, processor, OS...) and internet connection, I'll be able to help you setup OBS.