On what grounds are criminal laws challenged as to whether they meet Constitutional requirements?

There are a number of bases on which criminal laws can be
challenged.  It all depends upon what kind of conduct the law
prohibits.

For example, laws prohibiting flag desecration
are criminal laws.  They have been appealed on First Amendment, free speech grounds. 
Laws that ban various kinds of hate speech can be appealed on the same
grounds.

A criminal law can also be challenged on privacy
grounds.  For example, the laws that made it a crime to distribute birth control were
struck down on this basis.

It would also be possible to
challenge a criminal law based on the punishment it prescribes.  Such a challenge would
come under the 8th Amendment.

So the grounds on which one
can challenge a criminal law depend largely on what kind of action the law
criminalizes.