If we start heating a substance, does it melt when we break its intermolecular bonds?
I'm somewhat befuddled as to the confusion. I think it is perhaps the problem is in the phrase "intermolecular weak bonds."

If you look at a diamond crystal there aren't any diamond "molecules." Rather the whole crystal is one big "molecule." 

Likewise we write the chemical formula for sodium chloride as $\ce{NaCl}$, but in solid and liquid sodium chloride there are no sodium chloride molecules. True sodium chloride molecules can only exist in the gaseous state. 

If you look at $\ce{H2O}$ the hydrogen atoms are extremely labile. So the two hydrogen atoms and the one oxygen atom that were a molecule when the ice froze might not be part of the same molecule when the ice melts. 

So to fit the overall criteria of "intermolecular weak bonds," we need a molecule which has stronger intramolecular bonds than intermolecular bonds. Say methane for example. So when methane freezes the same molecule that freezes in the solid will be the same molecule that is freed when the solid methane melts. 

So in the case of methane adding energy in the form of of heat to solid methane will eventually reach the melting point of methane. At that point the weaker intermolecular bonds of the crystal break and the bulk solid turns into liquid methane.  

So the overall notion is that the solid form exists as colder  temperatures than the liquid form and that the liquid form exists at cold temperatures that the gaseous form. So generally we take solid and heat it. When (IF!) it turns to a liquid then the temperature of the transition is the melting point. Likewise if the liquid can be turned into a vapor then the temperature of the transition is the boiling point.  Thus solid methane and sodium chloride will both "melt," but for different reasons.