I plan on buying a PlayStation (possibly a PS4 pro) but I don't want to buy a TV and I don't own a TV.
Is there any way i can connect the PlayStation to my PC so that i can play the PlayStation games via the PC?
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No, you can't.
The VGA and HDMI ports on a laptop are for video out and not video in. Which means they cannot be used as a display monitor.
If you don't want to buy a TV, all you need is a cheap 20" monitor.
When you want to play PS4, connect HDMI cable from PS4 to the monitor. Vice versa, when you want to play games on your Alienware, connect HDMI cable from Alienware to the monitor. I am sure you will enjoy it more on the monitor than on laptop screen.
You can play the games remotely using PS4 Remote Play, however you will require to connect your PS4 to a display first in order to setup this feature. But once it's setup you can play streaming to your PC and thus not requiring a monitor. This comes with obviously all the downsides of streaming games (limited resolution and ocasional lag).
Now, I'm not sure the PS4 will not complaint about the lack of an HDMI cable being connected to the display port. I'm not sure, I have tried streaming and it works well enough but my PS4 is connected to a Screen while I do so, so you should give it a try. This might be as easy to solve as just plugging a HDMI cable to the PS4 while leaving the other end unplugged.
Additionally to options described in other answers, you can use video capture card. You can plug HDMI output of your PS4 in capture card, and use proper software (e.g. XSplit of OBS) to visualize input of your capture card. This usually works well for any device with HDMI output (e.g. any other consoles).