How to calculate the lift starting from the vertical speed? Is it possible to calculate the lift of an aircraft knowing its speed (IAS,TAS and GS), its vertical speed, its altitude and its weight?
Because if we look at the force of an aircraft in flight: the lift is opposed to the weight and if the lift is greater than the weight, the aircraft climbs with a certain rate (vertical speed) proportional to the lift.
Yes, it is, at least for small pitch angles and low thrust. Here is how:

$$L = m\cdot g\cdot n_z\cdot\cos\gamma$$

If thrust or pitch attitude become significant, you need to modify $m\cdot g$ by the vertical thrust component. Then the answer must be no, because the thrust information is missing:

$$L = \left(m\cdot g - T\cdot\sin(\Theta + \sigma)\right)\cdot n_z\cdot\cos\gamma$$

$L\:\:\:$  lift
$m\:\:$ mass
$g\:\:\:$ gravitational acceleration
$n_z\:\:$ load factor (= 1 in horizontal, unaccelerated flight) 
$\gamma\:\:\:$ flight path angle, with $\gamma = asin\left(\frac{v_z}{v}\right)$
$v_z\:\:$ vertical speed 
$v\:\:\:$ flight speed (TAS)
$T\:\:$ Thrust
$\Theta\:\:$ pitch angle (up is positive)
$\sigma\:\:$ thrust angle relative to airframe's x axis

Note that the altitude is only needed if you use the altitude-specific value of $g$. Using the standard value will give you a negligible error.

Please note also that lift > weight will result in an upward acceleration. In a quasi-stationary climb or sink the lift will be slightly smaller than the weight.