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I wanted to express my strong will and eagerness to achieve something (e.g 'I will destroy them',) So I came up with these 3 phrases : 'at all costs', 'no matter what happens' or 'in any case' to at the beginning of it. And I'm not sure if these 3 sentences mean the same or not.
At all costs, I will destroy them.
No matter what happens, I will destroy them.
In any case, I will destroy them
No, they do not all mean the same thing.
"At all costs" means that you will do anything to achieve your stated purpose or goal. Literally, you would pay any price to achieve it.
The most idiomatic ways of expressing your example would be:
They must be destroyed at all costs.
or:
I will destroy them, no matter the cost.
"No matter what happens" means that your statement is unconditional. It does not depend on circumstances and will go ahead in any possible conditions. Your example is perfectly idiomatic.
"In any case" is very similar to "no matter what happens" but usually refers to specific situations that you have already stated. For example, if you have considered various possible hypothetical situations but want to show that the outcome will still be the same you might state those cases and then say "in any case... [x] will be the result". Your example is okay, so long as it logically followed a statement of cases:
They may run, they may fight; in any case, I will destroy them.
