You do not want to ground to the electrical system. Ground yourself only to the computer case. That way you are always at the same potential as the circuit board whether its plugged in or not.

There is no need to be continuously in contact with ground. Just touch it before you start working and again if you walk away and return. A static charge must be generated, just standing or sitting there working will not generate any significant charge. It takes several thousand volts of static charge to do any damage, and those levels are not easy to accumulate. In very low humidity conditions, avoid wearing nylon clothing, it is an excellent static generator.

This whole business about static damage is highly overblown. I don't even own a wrist strap, never have. Just take sensible precautions and you will neve have a problem.