Closed loop control of PWM using current sensing method
I require 1A current constantly to the load despite change in duty
  cycle of PWM signal.


If you need 1 amp constant to the load, why are you trying to control it with PWM? Why not use a 1 amp constant current generator: -



The circuit above works by negative feedback to take enough current through the load to ensure that the voltage across the 1 ohm resistor equals the control voltage. So, if the control voltage is 1 volt, then the current through the load has to be 1 amp because the shunt resistor is 1 ohm. This can be made with smaller values of resistor such as 0.1 ohm but it can start to become trickier.

If you do need to control this with PWM (despite you wanting 1 amp constantly), you can low-pass filter the PWM output to produce the analogue control voltage. 


  Which method should I use to generate PWM waveform?


Do you need to is my question.


  What MOSFET should i use such that it withstands a little more than 1A
  current


Well, we don't normally respond to direct questions regarding product recommendations but most MOSFETs can handle over 1 amp especially DPAK, D2PAK, T0-220 case sizes etc..