3 pin 220 v us socket ground or neutral
The purpose of the 'earth' ground is to provide fault protection for you when you use your household appliances. By having one side of the incoming AC line connected to the 'earth', an internal appliance hot lead short to the case will cause a breaker to open or a fuse to blow - there would be sparks and a protective action.
If neither side of the line were grounded, the metal case of your toaster could develop a short to one side of the incoming line and you wouldn't know it. If your electric stove then happened to have a short to cabinet from the other side of the line, you would complete the circuit when you leaned against the stove to put toast in your toaster (you would get toasted instead of the bread).
The above reasoning is why metal military vehicles (ships, tanks, etc) take the opposite approach and use ungrounded systems. They don't want a fuse to blow or breaker to trip if one side of the AC line is shorted to the metal of the vehicle by battle damage. In this case the presence of a short is shown by 'ground detector' circuits that alert the operators.
In most developed countries, it is required that the ground pin of a three pin AC socket be connected to an independent ground wire that is returned to 'earth' outside the foundation of the building. This is done by burying a copper rod in the earth next to the foundation and connecting the socket grounds to that rod. Some jurisdictions permit using the metal water pipes of a building as a return path for the ground connection. There is possibility of galvanic damage to the water pipes from that sort of connection, so it is not generally the best way to go.
A few volts difference between neutral and 'ground' is normal. It is caused by the resistance and inductance of the wire that runs through the house to the AC point of entry, and by the resistance of the screw or clamp connections of the wires to socket terminals, breaker terminals, etc. There is no harm to your computer from the potential difference between neutral and the ground. Your computer case, like all the other appliances in your home, isn't connected to the neutral lead of the incoming AC line - it's connected to the ground pin on your three pin plug. The only time it has work to do is if a short develops in your computer power supply that connects the incoming hot line to the case - then a fuse or breaker should blow somewhere because of the current that will flow back through that ground connection.
As for adding a ground of your own - there is no harm in grounding the metal case of a computer or other appliance to a secondary ground connection. HOWEVER, if it seems necessary to do that when you already have a three pin grounded socket, there is something wrong with your socket ground. A qualified electrician should take a look at your circuit.