Convert Manual to Computerized "Automatic"

There is a manufacturer out on the market right now who actually does this. Magneti Marelli has an Automated Manual Transmission or AMT. This was discussed in this thread.

Yes in theory it could be done. But getting the mechanical components to work would be difficult and probably result in an awkward and bulky addition to your car. I suspect the end result would be an expensive and unreliable disappointment.

It would probably be easier to change the entire gearbox for automatic or a sequential box than try and adapt a manual.

Either way you're going to need access to a workshop with a lot of tools and machinery. You'll have to fabricate parts. And there would be lot of work finding parts that could move a gear shift and clutch the right speed the right distance with enough force.

If you're "not really a car" person I'm struggling to comprehend why you would want to do this.

It could be done...

but every action on a manual transmission takes a fairly hefty amount of force. So the mechanical actuators would be rather powerful. We're talking "Battle-bot" sized actuators.

Once you've got that, plus a few sensors, it would be a straightforward exercise.

I would give the system the ability to "bump" the throttle, so that after it has decoupled the clutch, it is able to spin the engine up to the correct, matching speed for the gear it is going into. This would greatly reduce both clutch and synchro wear. Note that this would require careful design so that a system malfunction cannot cause a throttle runaway event. It wouldn't need to have very much throttle authority, since it would only be revving the engine in idle. It's possible this could be accomplished with the existing Idle Air Control system, which is the system that keeps the engine at idle RPM despite additional A/C or power steering loads.