Streaming is extremely CPU intensive and you will not only need no less than i5 or higher, but at least 8GB or more RAM and a good graphics card. Even some higher end computers can have a hard time doing the process and it can come out very choppy.
As for lower end capture cards:
Dazzle DVC 100 or Hauppauge USB Live 2 as budget cards then something like these
through cables.
Second option for higher quality is either of the above or the Diamond Game Capture and this
s-video and rca cable.
You will run the standard RCA output to the CRT and then run the S-Video into the capture card. You will still need audio splitters with the first 2.
If you get the diamond you will just need another set of RCA cables because it has pass through and it will run out of the back and you need the cables to carry to the TV. It is also important to note you can use standard cables to input into the Diamond, you just run the Yellow cable into the green input and change a setting in the programming.
Programs:
I have not used the Diamond and am unsure of it's programming, but for the other 2 and if Diamond supports it I would use Amarec TV a program that interprets the video input of the capture cards incredibly fast using very little CPU. You can use this to record very easily. Even if your computer cannot stream, most can record using this and it will give you solid quality live recording. Also has many options to tinker with filters and video settings.
Then you will need a streaming program such as Xsplit or OBS. OBS is more hassle-free and you will use it to only stream, but import the video from Amarec and then add other cameras, text, layouts, etc. and then stream!
There are many guides out there this is merely a starting point of things to look at. Please do your research and be sure in what you want before you go too deep too fast.