Determine for sure how many speeds your shifter is. Clicks + 1 = "speeds". Most accurate method is to disconnect the shift cable, so that nothing interferes with the shifter. You can remove the rear wheel to count clicks, but you could still have a limit screw preventing you from getting all the "clicks"

1. Do you want to use your existing wheel ? If it's a 120 mm wheel, with 5 speed freewheel, jammed in a 126 mm frame, I think you will need a new wheel.
2. If you need a new wheel, then you can get one with a 6 or 7 compatible cassette hub. Or get one for a screw-on freewheel.