Partly, the "fake" chain is by definition the weakest one. The chains are built up block by block.
You could go back 100 blocks and start your new chain from there, you then only have to build 100 blocks. However, all the rest of the miners will be building on the main chain. So, by the time you have your 100 blocks done, they will have moved the main chain on. You need more than 50% of the hashing power to build your chain faster than the main chain.
The strength is down to 2 things. The main chain is growing fast and the further you want to go back, the more hashing you need to do just to catch up.
To completely replace a chain you need to re-do all the work already done, and grow your chain faster than the rest of the network.
However, if your chain did become the longest, then everyone will switch to it, and it will become the main chain.
A new alt chain wouldn't have much of a history to replace, but you would still need to grow the attacking chain faster than their main chain. This is a problem for alt chains. They generally don't have much hashing power growing their chain, so someone can easily overpower it. A bored ASIC miner could do it in many cases.