I recently bought an older home with a newer addition. The original
portion of the house has 2 conductor wire with no ground. The addition
and recently wired parts of the basement use 3 conductor wire and all
the outlets are grounded and appear to be wired correctly (when tested
with a $5 outlet tester from the hardware store).

I need to plug in some computer equipment to two older outlets in the
original part of the house. I understand that I need a "real" ground
wire back to the panel for surge protectors / computer equipment (no
GFC or ground tied to water pipes etc).

Here is what I want to do - I would like to know if there are any
concerns with this approach:

The new wiring for the basement and the addition are easily accessible
in the basement and each has a junction box located near the older
outlets I want to add ground to. I plan to run a ground wire from the
two older outlets where I need to connect my computer to the juction
boxes with the newer wiring in the basement, and connect them to the
ground wires in the juction boxes.

Assuming that the ground wires in the newer wiring are properly
connected back at the panel, will this present any sort of problem?
Will the two older outlets be properly grounded for use with computer
equipment?

Ben