Should the climax come after the MC achieves their goal at a high cost?
I believe you are misunderstanding what a climax is. To quote Wikipedia,
The climax (from the Greek word κλῖμαξ, meaning "staircase" and "ladder") or turning point of a narrative work is its point of highest tension and drama, or it is the time when the action starts during which the solution is given. (source)
That is to say, the climax is that part of the narrative when your MC starts their final run towards their goal. For example, the climax of The Lord of the Rings is when Aragorn has his army before the Black Gate, and Frodo is at Mount Doom. It is not after the Ring has already been cast into the fire.
'Climax' is often used as a synonym for 'orgasm'. It is useful to think of just that, to understand what a climax is: orgasm doesn't occur after one's finished having sex. It is the culmination of having sex. Similarly, the climax doesn't occur when the MC has already achieved his journey - it is the culmination of his journey.