So I want to buy a monitor with a USB-C dock in built like the one I have at work, but also want to make sure both my work computer and personal computer works on this.
Work computer is a Dell, personal is a Lenovo X1 Carbon. Is it as simple as any monitor with a USB-C dock will work or are there other things I need to look out for?
Broadly speaking yes it is that simple, USB-C PD (power delivery) is a universal standard and is largely just plug and play. Of course it's IT so nothing is ever quite that straightforward, you need to ensure the monitor can deliver the same wattage of power as what your laptop needs but this is mostly going to impact either very cheap monitors or very expensive laptops designed for gaming.
Your work Dell and Lenovo X1 should both probably need 45-60W max from your monitor in order to stay charged and run efficiently, so just ensure the monitor you buy has that same capability. There's a few on the market that only do 15W for example which are designed more for charging a smartphone than a laptop, and they won't keep your laptop charged.
You can't go wrong buying the same monitor as the one you have at work given you know it already works and this is what we recommend to our staff. I buy Dell monitors for our workplace if it helps, we've had Dell P3421W and more recently C3422WE conferencing monitors. But Dell offer a wide variety of 24, 27, 32 and 34" monitors with USB-C PD.
My final piece of advice is don't cheap out on the USB-C cable; make sure it's a modern SuperSpeed cable ideally; much like newer HDMI cables these are much higher bandwidth (10gbps or 20gbps). It's not going to make or break your setup necessarily but as people start plugging in more and more peripherals, sit on video conferences all day etc that bandwidth can help. We've had lots of issues with staff complaining about peripherals stopping working or their calls dropping and either upgrading the cable or adding a second USB superspeed to the config almost always alleviates it.