Step 1: Identify the intermolecular forces present in each hydride.
HCl: Dipole-dipole interactions, London dispersion forces.
NH$_3$: Dipole-dipole interactions, London dispersion forces, hydrogen bonding.
H$_2$O: Dipole-dipole interactions, London dispersion forces, strong hydrogen bonding.
HF: Dipole-dipole interactions, London dispersion forces, very strong hydrogen bonding.
Step 2: Compare the strengths of the intermolecular forces.
Hydrogen bonding is significantly stronger than dipole-dipole interactions and London dispersion forces. The strength of hydrogen bonding depends on the electronegativity of the atom bonded to hydrogen and the presence of lone pairs. Fluorine is the most electronegative, followed by oxygen and then nitrogen.
Step 3: Predict the boiling points based on the strength of intermolecular forces.
Stronger intermolecular forces lead to higher boiling points.
HCl has the weakest intermolecular forces among these, so it will have the lowest boiling point.
NH$_3$ has hydrogen bonding, making its boiling point higher than HCl.
HF has very strong hydrogen bonding, leading to a significantly higher boiling point than HCl and NH$_3$.
H$_2$O has strong and extensive hydrogen bonding, resulting in the highest boiling point.
Step 4: Arrange the hydrides in increasing order of boiling points.
The increasing order of boiling points is: HCl<NH$_3$<HF<H$_2$O.
Final Answer: The final answer is $\boxed{HCl<NH_3<HF<H_2O}$