What would happen if you tried to follow two Bitcoin blockchains from the same computer? This will probably sound stupid but what would happen if you create an entirely new Bitcoin block chain and run it on the same computer. How does a node know which version of the Bitcoin block chain to talk to? Does each block chain have a unique identifier?
Bitcoin is programmed to accept the most proof-of-work chain, which is many times also the longest chain. Every new block added to the blockchain references the previous block with its hash. Based on your question, there are a number of cases that I have come up with and how your new 'forked' blockchain will react.


You run the same software that Bitcoin is currently working on: your software will automatically accept blocks from other peers and add it to the genesis block. This will bring you to the current Bitcoin blockchain
You change the genesis block, but use the same DNS seeding: when the peers will send you new blocks, the previous block hash reference in block 1 will not match your new genesis block header hash and hence that block cannot be added. This you have now generated a completely new chain.
You fork the blockchain but change rules at certain point: This is how Bitcoin Cash was created. From a particular block height they increased the block size to more than 1MB, which resulted in the original Bitcoin nodes rejecting the newly generated blocks. So you now have a different chain from a particular height.