The thing is, shiny Pokemon's chances are calculated by IVs. So, in order to get a shiny Pokemon, you have to have the following:
In Generation II, being Shiny is determined by a Pokémon's IVs. If a Pokémon's Speed, Defense, and Special IVs are all 10, and its Attack IV is 2, 3, 6, 7, 10, 11, 14 or 15, it will be Shiny. Because of this, a Shiny Pokémon traded to a Generation I game and then traded back to Generation II will retain its Shininess, and a Pokémon obtained in Generation I whose IVs meet the requirements for Shininess will also become Shiny when traded to Generation II.
Since the HP IV is calculated from the other four IVs, a Shiny Pokémon's HP IV will always be either 0 or 8. Shiny Pokémon are generally above average in terms of IVs, but only slightly.
That, however, only relates to catching shiny Pokemon. Here are the breeding mechanics:
The probability differs for bred Pokémon, as their IVs are partially influenced by their parents. Specifically, a parent passes its Special stat (plus or minus 8) and its Defense stat to its children that are the opposite gender from it. (If Ditto is one of the parents, it is always the one that passes these stats.) This means that if a Shiny parent passes its IVs to a child, the child's inherited Defense IV will always be 10, its inherited Special IV will have a 1/2 chance of being 10, its randomly generated Attack IV will have a 1/2 chance of being an appropriate value, and its randomly generated Speed IV will have 1/16 chance of being 10. This results in a 1/64 chance that such a child will be Shiny. These breeding mechanics also mean that if a child inherits its IVs from a Pokémon that does not have a Defense IV of 10 and a Special IV of 2 or 10, it cannot be Shiny.
From all of this, you have a 1/64 chance of getting a shiny Pokemon.