I keep my BTC in Bitcoin Core wallet. I would like to start using my BCC but full node is downloading whole blockchain. How can I avoid it?
If you want to support BCC without downloading blockchain, there are two options.
1) Bitcoin ABC + pruning
Pruning allows your Bitcoin-Qt/BU/ABC to work as a node without downloading the entire blockchain.
See this on how to prune: Can Bitcoin-Qt be configured to trim the blockchain?
2) Electron Cash
Electron Cash is the changed version of Electrum SPV client with BCC support. As it's a SPV client, ----it won't download the entire blockchain-----, as change suggested by @Pieter Wuille, SPV clients do download the blockchain, they just don't store entire blocks, they keep only the required parts.