I want to analyze transistor transient characteristics. What should I do first?
Q1 works as a switch. The gate should be connected to ground with a fairly large value resistor (10k-100k), so when you turn of the switch the gate charge will dissipate and close the transistor.

Now. About the BJT. It all depends on how accurate you want to get. Basically when the switch is on, R4 and R5 form form a voltage divider charging C1 to 2.5V (half of the voltage of B1). This is a simple example of an RC Vc = Vin * (1 - e^(-t/RC)), or in this particular case the Vin = 2.5V and R = R4 || R5. 

Since the load is in the emitter, the circuit acts as an emitter follower, the emitter voltage will be about 0.6V lower than that on the base. If base voltage is below 0.6V you may assume the laser to be off. Above that the voltage on the emitter will be Ve = 2.5*(1-e^(-t/(5e3*10e-12)))-0.6. The voltage on your laser will depend on voltage drop on R6 and the current flowing through it. 

However, this is a rather poor circuit, the maximum voltage on Ve will be around 1.9V, meaning that 60% of the power will be dissipated on the transistor (the rest on R6 and the laser). The circuit would perform much better R5 would be removed. Then the emitter would go all the way to about 4.4V. 

All these calculations assume that the transistor Beta is rather large and the base current is not that significant to influence the RC constant.