Why is naphthalene less stable than benzene according to per benzene ring?
A short answer is: you're forgetting that a double bond is shared between the two rings in naphthalene.
A long answer is given below.

In terms of heat of hydrogenation

Naphthalene can be hydrogenated to give tetralin.
Further hydrogenation gives decalin.



Scheme 1: hydrogenation of naphthalene.

On the other hand, the hydrogenation of benzene gives cyclohexane.



Scheme 2: hydrogenation of benzene.

In an old report it reads (Sherman, J. Oil Soap (1939) 16: 28):


  If benzene could be simply regarded as cyclohexatriene with no interaction between the double bonds, the heat of hydrogenation to form cyclohexane should be just three times the heat of hydrogenation of cyclohexene to form cyclohexane.
  From Table I, this is $3 \times -28.6 = -85.8$ kcal/mol.
  The experimental value is $-49.8$ kcal/mol.
  Hence benzene is more stable by an amount $-49.8 -(-85.8) = 36.0$ kcal/mol than it would be if it were completely unsaturated in character (with no interaction between the double bonds).
  From the theoretical viewpoint, this extra stability of benzene (and other aromatic compounds) is shown to be a consequence of the fact that the normal state of the molecule is not that corresponding to either Kekulé structure but is a sort of combination of the two.


This $36.0$ kcal/mol may be regarded as a direct measure of aromaticity and is cited in many places as resonance energy.

In particular, the resonance energy for naphthalene is $61$ kcal/mol.
This value is shifted by around $5 \times -28.6 = -143.0$ kcal/mol (five double bonds) from the actual heat of hydrogenation of naphthalene.
This means the heat of hydrogenation of naphthalene would be somewhere around $61 - (-143.0) = -82$ kcal/mol.
The actual value from NIST (to cis-decalin) is $-318$ kJ/mol, or $-76$ kcal/mol.

So it costs $-49.8$ kcal/mol to hydrogenate benzene to cyclohexane but only $-76$ kcal/mol to hydrogenate naphthalene to cis-decalin, less than twice a benzene.
Furthermore, part of this energy is due to the resonance energy, which is $36.0$ kcal/mol for benzene, but only $61$ kcal/mol for naphthalene, again less than twice a benzene.

What did we forget?

Naphthalene rings are fused, that is, a double bond is shared between two rings.
This makes the above comparisons unfair.

A better comparison would be the amounts of resonance energy per $\pi$ electron.
Benzene has 6 $\pi$ electrons, which makes its resonance energy equal to $36.0 / 6 = 6$ kcal/mol per $\pi$ electron.
Naphthalene (10 $\pi$ electrons) shows a remarkably similar value: $61 / 10 = 6.1$ kcal/mol.

Even comparison of heats of hydrogenation per double bond makes good numbers.
For benzene (hypothetically three double bonds) it costs $49.8 / 3 = 16.6$ kcal/mol to hydrogenate each double bond.
For naphthalene it would be $-76 / 5 = 15.2$ kcal/mol, again very similar.

What about tetralin?

Again NIST comes to our rescue.
The most recent value for the heat of hydrogenation from naphthalene to tetralin is $-125$ kJ/mol, or $-29.9$ kcal/mol (Scheme 1, first arrow).

As a per double bond value it gives us $-29.9 / 2 = -15.0$ kcal/mol, close to both benzene and naphthalene (all of them differ by less than $1.6$ kcal/mol).

In terms of electronic structure


  Both are aromatic in nature both have delocalised electrons but naphthalene has more number of $\pi$ bonds and hence more resonance structures and more delocalisation so overall it must be more stable.


I believe the highlighted sentence tells it all.
As you said, delocalisation is more significative in naphthalene.
A simple model for delocalisation is the particle in a box (in this case, more of a rectangle).

The $n$-th energy level for an electron confined in a single dimensional box (line segment) is

$$E_n = \frac{n^2 h^2} {8 m_e L^2}\text{,}$$

where $m_e$ is the mass of the electron, $h$ is the Planck's constant and $L$ is the length of the line segment.
So energy decreases with the square of the length of the confinement.
Results are analogous for other dimensions.

From this simple model, the more confined an electron, the higher will be its energy.
It is thus perfectly reasonable to rationalise that more delocalisation results in more stability.
Given its larger delocalisation, it seems rational that the energy levels of naphthalene have less energy in comparison to benzene.