clutch problem?

If you squeeze in the clutch lever and the rear tire still wants to turn whenever the engine is cranked over then yes, you have clutch issues.
If you could describe in more detail just exactly what the issues you are experiencing are while "turning on" your bike, I and many other knowledgeable folks here would be glad to help you get to the bottom of the problem.

I think you need centrifugal clutch with that pull start. With a friction clutch, you will experience what you have now.

Or you can just lift the rear and pull start it, to get it started.

The pull start turns the crank shaft only. If he squeezes in the clutch it disengages the crankshaft from the engine sprocket through the operation of the clutch. The pull start then would only rotate the crankshaft and the bicycle would not move forward.
If the bicycle does indeed move forward with the clutch in the disengaged position (hand lever squeezed in) then there is a problem with the clutch.

Pull starts are recommended to be used along with a centrifugal clutch because a centrifugal clutch does not provide a method of starting the engine through pedaling. (some claim to but are not reliable)

How tight do you have the clutch cable adjusted?
The most common error I have seen is a clutch cable with too much slack in it.
With the cable adjusted properly, hand lever released all the way out, you should barely be able to wiggle the cable about 1/16" to 1/8" and the release arm should not be able to pivot inwards towards the carburetor easily at all. The release arm should be able to be jiggled a teeny tiny bit though. You do not want alot of pressure on it as that will load the release mechanism as you are riding down the road and lead to accelerated wear on all the release components.