Nothing is really impossible, but is very limited, the best you could do is EV train him, it will help a little but you
will still have a bad time against many enemies. Not because his overal stats but most because of his movesets.
But if you fight against Pokemons who are weak against bug bite, you may have a little fun, but will be a bit difficult. One thing that I learn, is that you don't have to have the perfect Pokemon to be good. If you only play with brute force, this may be difficult, but most pro players use strategy over brute force.
so if you have the Pokemons to cover you caterpie weakness. You can win alot, I just can't say that you will only win with the caterpie alone, but you can have fun using the catterpie against some.