How does the Blockchain detect a cheater, who manipulates the blockchain? Let's assume we have 100 blocks. One miner x has the intention to manipulate block 90. So he has to recalculate the hash for block 90. Following he also has to recalculate the hashes for block 91-100. 
Correct me, if I'm talking non-sense here.

So how do all the other miners figure out that miner x is cheating? And what happens to miner x, if he gets caught?
What you are describing is a textbook example of a 51% attack. Bitcoin protocol is designed to accept the chain that has the most proof of work (most often this is same as the longest chain). If a miner needs to change block 90, he will have to recalculate hashes starting from block 90-100. To make his attack successful, the miner will have to mine blocks faster than the rest of the network so that he can broadcast the longest chain which then gets accepted by the network. So, probabilistically this would mean the miner should be in control of >50% of the total hashing power, so that he can mine blocks faster than rest of the network combined.