Technically the miner programs can mine both ZEC and ZCL. You can schedule your mining on one or the other as you want provided you have enough CPU cores or GPUs to handle the desired load and balance you want to achieve. Some pools (I mine on suprnova.cc) allow you to maintain the same workers / passwords. This allows you to flexibly decide when you want to mine, with which workers and on which coin/pool.
You still need to have the wallets operate side-by-side as ZCL is a fork of ZEC and as @thesmokingman wrote above they are on different chains. I was writing recently about how I managed to have the ZClassic and ZCash wallets operate side by side on my mac. The procedure is the same. Link is here: ZCash Desktop GUI Wallet - #122 by neuromancer (see posts 121 to 123).
Good luck and have fun!