(1)  Make a name for oneself = become famous

• Balabhaskar made a name for himself as a talented violinist.
• By the time she was thirty Lagi Varkey made a name for herself as a successful Headmistress.
• By hard work and determination James made a name for himself and became the best trainer in Engineer’s Driving School.

(2)  On one’s way up = rising to better and better position.

• John is definitely on his way up in the financial world.
• He started as a junior officer and worked his way up to become an AGM in the Bank.
• She quickly worked her way up to the position of a senior vice-president.