You can do it. However, Bitcoin Core's default directory (Windows) is %appdata%/Bitcoin so you will have to specify the data directory to be in your external hard drive instead of the hard drive that you installed Core on. You should be able to specify it when you first start the client, I'm not sure whether the settings will be saved however.
Take note that your wallet and blockchain will be at wherever you specified it to be.
i think use online wallet is better than install on hard drive .
for example blockchain.info
Please refrain from commenting on anything that you don't know about. OP SPECIFICALLY asked whether it is possible to install Core on a hard disk. Spam elsewhere if you are not going to answer OP's question.