Steering wheel can be shaking for several reasons. Your tires may be worn or out of balance. You can have several front end parts worn. With 166000 miles this is very possible. Also, your brakes rotors could be warped. Front end could also be out of alignment. My suggestion is most front end shops provide free inspection to tell you. You don't want to drive it if a front end part is bad.

I have also provided a pic for you to show you all of the possible parts on the front end.......also - your shocks could be bad. Again, please have checked out.

Thanks ImpalaSS, I forgot to mention that I bought four new tires and had a wheel alignment. I was thinking that it's the CV joint.

Could the CV joint cause my steering wheel to shake when I hit 60 mph?

Again, it is possible......there are so many front end parts that can go wrong. Without the ability to look at them I would be pointing you in the wrong direction and don’t want to do that. To me the best thing is to take it to two different front end places for the free look at estimate. This way you can compare what they both say and pretty well zoom in on the right problem. Tie rods can be loose and allowing the shake but again, I don't want to point in any one direction. As you can see from the front end parts.....there are many. The shocks could also be bad and allowing the tire to hop.