Basically, if you want to get any character besides a Universal, you need to just give decent care. Which means raising its skill points to a decent level, maybe around 70-100 (haven't really experimented), filling up hunger and happiness hearts as soon as one empties (well you can let more drop just as long as you don't starve it too often) and training may also be a factor, though it doesn't seem to influence growth too much. Also, keep your Tamagotchi's weight down as much as possible.
On the v4, for the most part, what character you get is not influenced by care. You can just treat your Tamagotchi decently and get a Mametchi, or a Memetchi, or a Kuchipatchi, or a Tarakotchi. The two things that really affect growth on this Tamagotchi are what teenage character you get and what skill has been raised the highest. You could have 200 Intelligence skill points and still not get Mametchi if you didn't get a right teenager or if you raised Style or Kindness too much.
As far as I know, what teenager you end up with is just luck. I find that the Meme family is the most abundant and the other two families are much less common.
So, if you want a Mametchi: you will need to get a Yangu Mametchi or a Yangu Robotchi/Young Androtchi. Then, you will need to make sure that its Intelligence points are the highest.
For Memetchi, you'll need either a Yangu Memetchi or an Ichigotchi. Then you'll need to make sure that its Style points are the highest.
For Kuchipatchi, you'll need either an Oniotchi/Oniontchi or a Yangu Kuchipatchi. Then you should raise its Kindness points to be the highest.
For the secret characters (Tensaitchi, Makiko and Nonbiritchi) you will need to obtain a certain character, then raise its skill points to a total of at least 450. At least 350 of those skill points must come from a certain stat.
Tensaitchi evolves from Mametchi. It needs 450 skill points total, at least 350 coming from the Intelligence category.
Makiko evolves from Memetchi. It needs 450 skill points total, at least 350 coming from the Style category.
Nonbiritchi evolves from Kuchipatchi. It needs 450 skill points total, at least 350 coming from the Kindness category.