You need a combination of

gvusb2 - https://www.amazon.com/DATA-connection-video-capture-GV-USB2/dp/B00428BF1Y

and

rca splitters - https://www.amazon.com/iExcell-Speaker-Splitter-Shielding-Adapter/dp/B019ZQS4N4/ref=sr_1_3?dchild=1&keywords=rca+splitter&qid=1602468047&s=electronics&sr=1-3

You'll split the audio (red/white) and video (yellow) signal from your NES to go to both the TV and your computer via the GV-USB2 (or equivalent device)

You CAN show the display on your computer only... but it will have horrendous unplayable lag. The video going to your computer will be for the purposes of showing on OBS only.

I'll make an addendum to what Darpey said.

Since you said you have to use the RF modulator, I'm assuming the TV does not have Composite (RCA) inputs. If that's the case or in the instance you don't mind the quality on your TV you can do it this way:

Still continue to run RF to your TV and use the Composite (RCA) jacks to the capture card. The NES can handle both fine simultaneously. I did it for 7 years. In this scenario, you're not required to get a different TV and you don't have to buy a splitter if you don't want to. The only equipment you'd absolutely have to buy would be the capture card.

Agreeing with Darpey, you can just play on your monitor with a capture card, but you have the added delay of modern screens plus the delay of the capture card processing the video on top of it. I'm pretty certain you'd have a rather difficult time adjusting to that.

Does your video card have RCA female connectors? If so, you don't need a capture card. If not, you plug the cables into the capture card, and connect the capture card to your computer via USB (or HDMI potentially). Purpose of the Capture card is to deliver (and sometimes upscale) your NES video/Audio signal to your PC.

You need a capture card to convert the analog RCA signal to a digital USB/HDMI signal. Sounds like your best setup would be RF to TV and RCA to capture card to PC as @roopert83 describes above. You don't need a splitter with the way you're describing your setup. If you want two RCA output signals, then you'll need the splitter.