The reason he knows he has done a bad thing which he must cover up is because he has been through this before. He has accidentally killed things due to his inability to understand his own strength, control himself, etc. He knows he has done a bad thing based on the past reactions of George and others. This is why he then covers it up and hides the deed from Curley's wife nd George.
Lenny knows he did something wrong because he had the pup covered in the hay to hide it from George. He kept talking to himself about how he didn't want to get caught because he might not get to tend the rabbits if he gets caught doing anything bad. He covers the pup when Curley's wife came in to talk to him trying to hide the bad thing he's done. He also felt the need to explain when she asked what he had covered up. Lenny feels badly about it, but he also feels it was the puppy's fault for playing like he was going to bite Lenny.
It isn't that Lenny is really aware of the severity of the dead puppy as much as he is aware that if George finds out he might not get to tend the rabbits. He doesn't really understand fully the finality of death yet.