Is it normal to think about death often?

I've gone through several deaths in the family in recent years and I've been thinking that one day I'll die. The thing is, once I start thinking about that, I'll get this feeling like what will happen to me when I die? I know my body will rot but what about my soul?

I also have a kitten who I believe is dying. He has green film in his eyes and can't eat meat. He's also lost weight and comes near me. The only I can do is pet him (I wash my hands afterwards) and tell him he's in God's hands.