How to calculate the lift starting from the vertical speed?
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.