I am in the process of renovating a bathroom in my basement. There was only a shower and a stool. I want to add a sink. There is a drain pipe with a cleanout from the kitchen sink that comes right down along the wall in the bathroom. Can I take out the cleanout plug and tap into it for the drain for the sink I want to add?

Can I take out the cleanout plug and tap into it for the drain for the sink I want to add? No! But what you can do is cut in a sanitary tee 18" to center above the floor to pick up the drain. But you will have to vent your sink, Come off the first tee with another laying on its back.
You may now run a vent up i8nto the attic and revent back to a existing roof vent or simply add a AAV on the branch of the tee, Good luck Tom