A Memoir Of A Dog (this title doesn't really make it obvious that the story is written from a dog's POV, maybe something like "Growing Up Dog - A Memoir by Brown Egg")

I have seen many other people depicting their lovely pets in their notebook entries, which makes me a bit jealous. I didn’t have the honor to be put my cute picture under my host’s masterpiece. Anyway, my elder sister Echo is a slob(I hope she won’t lose her temper upon noticing this) and she has nothing to be called masterpiece at all (nothing she writes can be considered a masterpiece). I guess she considers other things more important than this, so I decided to write a short bibliography about myself to show you what my life is like.

“Brown Egg”, that is my name. Five years and six months ago, I was brought into this world. I scarcely can recall days of my childhood, not because of my short memory, but because I tried to avoid the days I had been teased by others. People used to tease me when I was young. You know, some doggies are not that pretty when they are still little puppies. Unfortunately, I was one of them. I was told once in the cradle by my step mother (the way I call my previous host before I was adopted into my present family) that I looked like a rat. Jesus! Do I really look like a rat? Of course I don’t look it that way, but I have been aware of my homely appearance ever since then. I know some reasons somehow. Maybe it is because I was not born in a decent way (come from common stock), I mean, I belong to no rare breed. My mother who gave birth to me was a Chihuahua, while my father was a….nobody. Perhaps he was of half Pomeranian and half Papillion. Or Otherwise, people wouldn't have called me like a “freak” in the early days of my life.

Luckily, I was adopted as an outcast by my present family 2 months after I was born. Gradually I got to know they happened to come by over a relative that day, and this is the way that was intended. (200 words omitted here)  (I don't understand what you're saying here) There would have been lots of misfortune that might have happened to a poor doggy, but I was a child of fortune, perhaps the luckiest one.

Good story =) 

Dark green is additions I have made that make it sound better
Yellow is another way of phrasing your original sentence in a clearer or more effective way
Red is a correction of grammatical mistake
Dark red is something I don't know how to fix :P