How is Lennie's treatment of his dead puppy typical of his character?
One of the first instances of Lennie's character occurs in chapter 1 when we see how he treats the dead mouse. He doesn't seem to really care that the mouse is even dead, but he cares to be able to pet soft things. We can assume the same is true in the dress of the woman that he was touching when they were in Weed and got chased out because of that.
Death doesn't really register to Lennie as significant. He doesn't even care that his puppy is gone, he cares that he's going to get in trouble.
This shows a couple of things about Lennie's character worth noting. First, it demonstrates that Lennie has little concept of relationship, animal, or the significance of life - it can be snuffed out and gone in an instant. Second, if these traits are typical, they might foreshadow the climax of the story. In most literature, when character traits are revealed and repeated, they are done so for a reason, watch for the coming reason of being able to identify the trait of Lennie's facination for soft things.
To me, what his treatment of the puppy shows is that he is both very gentle and very prone to sudden bursts of rage that are really sort of irrational. In addition, it shows that he is not really capable of deciding things for himself -- of knowing what is right and what is wrong.
You can see his gentleness as he strokes the puppy and mourns the fact that it is dead. You can see his rage (the sort of rage that leads him to kill Curley's wife) when he flings the puppy away from him. You can see that he is really not clear on what's right and what's wrong when he first wants to hide the puppy from George and then starts thinking that maybe what he did was not really bad.
The only thing that I would add is the important decision that Lennie makes in part five: he has two dead bodies with him. He chooses to half-bury Curley's wife in the hay, doing such a poor job that Candy immediately sees the body when he enters the barn. He even scolds her for sleeping in there. Lennie, thinking that both deaths are "bad things" for which he will get punished, and viewing them as equal, hides the body of the puppy instead.
It is typical of his character to not realize the consequences of his actions, besides assuming that, as a result, he won't get to tend the rabbits.