Correct spelling, explanation: achieve is based on the French word achever. The spelling and pronunciation have adapted to English over time. Most notably, che became chie, because English strives for the most comfortable way to pronounce words. The ending er also changed to fit English spelling, but the structure of the word generally stayed recognizable.

Definition of achieve:
verb
1. to succeed in doing or becoming something or fulfilling an ambitious plan.
He’s very young but he won a lot of competitions, so I’m sure he’ll achieve even more in the future.
Steve is working hard to achieve his goal of becoming a doctor.

Incorrect spelling, explanation: while acheive may seem not too far off the correct word – achieve – you need to consider how it would be pronounced. The diphthong ie and ei have completely different pronunciations, because of which acheive no longer sounds similar to the original French achever.

Incorrect spelling