In The Crisis, what does Thomas Paine mean when he states, "I cannot see on what grounds the K\king of Britain can look up to heaven for help...

When Thomas Paine wrote the quote that you have presented here, he meant that the King of England did not have the moral high ground in the dispute with the colonists.  He meant that there was, therefore, no reason that God would favor the King over the colonists.  We can see this from the rest of the sentence from which your quote comes.

In those days, most people believed in God and most people believed...

When Thomas Paine wrote the quote that you have presented here, he meant that the King of England did not have the moral high ground in the dispute with the colonists.  He meant that there was, therefore, no reason that God would favor the King over the colonists.  We can see this from the rest of the sentence from which your quote comes.

In those days, most people believed in God and most people believed that God would take sides in disputes between countries or groups of people.  In the third paragraph of Chapter I of The Crisis, Paine says that he believes that God is in charge of the world.  He says that he does not have

so much of the infidel in me, as to suppose that He has relinquished the government of the world, and given us up to the care of devils…

This shows us that Paine believes that God will side with either the colonists or the King in the dispute between the two.

Since he believes this, Paine wants to think about which side God will take in this dispute.  Undoubtedly, God would take the side of the good, moral people.  Paine argues that the King is not good or moral.  He implies that the King is, instead, no better than a common criminal.  He says that the King has no more cause to expect God’s help than “a common murderer, a highwayman, or a house-breaker” would.

So, in your quote, Paine is saying that the King is not moral.  Because the King is not moral, he has no reason to believe that God will help him against the colonists.  Because the King is no better than a common criminal, he has no grounds (reason) to look up to Heaven (God) for help against the colonists.