How to run a private bitcoin network with low difficulty?
Use regtest

Chain selection options:

  -regtest
       Enter regression test mode, which uses a special chain in which blocks
       can be solved instantly. This is intended for regression testing
       tools and app development.

  -testnet
       Use the test chain


Using this mode, you can manually mine blocks by issuing the following command, where nBlocks is the number of blocks to mine. You will need atleast 101 blocks before your first mining reward becomes available.

bitcoin-cli generate nBlock