First of all, I assume you know that the creature does kill himself (or at least goes off saying he will kill himself) at the end fo the book. So I assume you mean why didn't he just kill himself earlier -- why kill all those other people.
If that's what you're asking, then the answer is that he wanted to get revenge on Frankenstein. After all, it was Frankenstein's fault that he was in this predicament. Frankenstein made him mentally and emotionally human, but physically hideous.
So, the creature wanted to get revenge. It was only when he saw Frankenstein dead that he really regretted what he'd done. That's why he kills himself when he does.
In Shelley's Frankenstein, two motivations are present that keep the monster wanting to live.
First, he hopes for a female mate. He believes this will make his life bearable. He will have someone as a companion that will not be appalled by him, since she will look something like him.
Second, once Victor impulsively destroys what he has created of the female, revenge is the monster's motivation. He plays cat and mouse games with Victor, leading him on and humiliating him once in awhile just for fun.
First, then, hope for companionship keeps him going, and then revenge takes the place of hope.
Survival is an innate human drive. If not, many more people would take their own lives when pressure arrives. Victor's creature spends his first part of his existence longing for Victor and trying to figure out what his life is about. He hides and he learns. He becomes an educated person.
By watching the family and their goodness, the creature develops a sense of hope. He eventually begs them allow him to be part of their lives when he confronts the blind man. Hope for a connection to another human has helped motivate him to survive.
He again has hope when he demands a female creature to be made for a mate. Victor crashes his hope and the creature recognizes that he will be isolated from mankind.
He replaces hope with anger and begins to seek revenge on Victor and society for rejecting him. Victor has abandoned him so he seeks his revenge in payback of Victor's abandonment.
When the creature places himself in the frozen wasteland, we know very little of what things he ahs done to others who live in the wilderness. We know he has a sled of dogs so the reader may surmise that he took them and could have harmed someone while doing that act.
The creature lives because his last hope is that one day he will be able to connect with his father, Victor. He needs the connection and waits for it. Victor follows him to the ends of the earth with intent to destroy the creature, but that sense of hope still exists for the creature.
In the end Victor perishes and it is then that the creature has lost all hope and a desire to live.