Remove the Handle
Its a good idea to put some newspaper under the door as it is very possible that metal dust will fall on the floor as the handle is removed. It can also be a good idea to run a sharp knife around the edge of the handle if it was there the last time the door was painted.
Removing the Spring
For this you will need to protect your eyes and have a circlip pliers. Use these to remove the circlip from the groove. Now you will need to remove the washers and the cam. Make a note of the orientation of the cam as you remove it. Now move the handle up to remove any pieces of the last broken spring to add the new one. Some grease or vaseline here can help reduce wear on the new spring.
Make sure to use circlip pliers to remove the circlips as attempts to remove by prying them will likely deform them. Also make sure to wear appropriate eye protection as circlips can occasionally come flying off the pliers.