For windows there is PSPICE and ORCAD. Not sure what is available on the linux side.
How do I connect my circuit with computer
Kind of an open question. We need a lot more info.
What is your circuit?
What do you want the computer to do with your circuit or the circuit with the computer?
The two common interfaces found on a computer is a RS-232 serial port and the parallel port also known as the printer port. There is a ton of information you can find by searching the internet.