Sorry if some of this info is wrong, I havn't seen much of Trunks in action.
As the poem on the main Trunks page indicates, Trunks did not have a bright and cheery lifestyle. He lived during a time when Jinzouningen Juunana-Gou and Juuhachi-Gou (Androids #17 & #18) literally trashed the Earth. The strongest of Earth's fighters, that meaning Tenshinhan, Yamucha, Piccolo, Kllyln, Chaozu, and even Trunks' father Vejiita died trying to save the world. The Z fighters as they were called, tried their best but failed. Son Gokou, the main character in DBZ, died of a rare heart disease and not in battle.
Trunks grew up never knowing what his father was like besides that of what his mother, Bulma, told him about Vejiita. The only other company Trunks had besides Bulma was Son Gohan, Gokou's son. Trunks liked Son Gohan a lot, for Son Gohan was not only his best friend but also his mentor and teacher when it came to fighting. Trunks and Son Gohan were the Earth's last hope.
Trunks was 13 years old when he witnessed the battles Gohan waged against the Androids. Little did he know of what fate will soon come to his valued friend. In the final battle Son Gohan fought without Trunks to watch. The Jinzouningen had the upper-hand and defeated Son Gohan with a bombardment of their increadible power. Later on Trunks found Gohan's body and it was when he felt hurt and angry over his friend's death did he first become a Super-Saiyajin.
From then on Trunks continued to train himself and improve on his strength, determined to beat the killers who have taken the lives of millions. In the meantime Bulma created a great invention. She invented a one-person time machine so that Trunks could travel back to the past, warn the Z fighters of the upcoming threat, and also give medicine to Son Gokou for his heart condition.
When Trunks arrived in the past he easily defeated Freeza and Freeza's father, King Cold. He does in fact complete his mission and he also learns about his father as well. Vejiita was a full-blooded Saiyajin as well as the Prince of the Saiyajin race, so that made him extremely stubborn and arrogant. He ranked his pride above practically everything even when he became a good guy. Even under such a shell, there is a tint of emotion in him that perhaps only Bulma coud fully grasp.
Once he knew that the future would be safe, he returned to his timeline to tell his mother. However, in the space-time continuum no event can be altered entirely once it happens. That means that Trunks' world stays the way it is, but the past that he altered would have a different future and be its own realm. This information wasn't all too pleasing for Trunks, but he knew that he did the right thing. He was the hope for the future and he fulfilled his destiny.
