Melting and boiling are changes of state. They are physical changes, so no new substance forms and no covalent bonds break in a simple molecule. To melt or boil a substance, its particles must gain enough energy to overcome the forces holding them together. The stronger the forces, the more energy is needed, and the higher the melting point or boiling point.
The forces that must be overcome depend on the structure. In simple molecular substances such as methane or water, the covalent bonds inside each molecule are strong, but the intermolecular forces between molecules are weak. Only these weak forces are overcome, so little energy is needed and melting and boiling points are low. Larger molecules usually have stronger intermolecular forces, so they need more energy to boil and have a higher boiling point.
In giant structures such as diamond, sodium chloride or a metal, strong bonds act throughout the whole structure. A lot of energy is needed to overcome them, so melting points are very high.
While a substance is changing state the temperature stays constant. The energy supplied is used to overcome the forces between particles, not to increase their kinetic energy.