Yes, you can download apps from the Google's Play store directly on your PC, but you will need an additional tool and, of course, you won't be able to install the application that you download on your PC.
There's a program called APK Downloader, which allows you to do just that. As far as I know, it can work as an extension for Chrome as well as Firefox. The tool will ask for your Google account and your phone's ID before starting to download apps. In case you don't know your Android device's ID, there's an app called Device ID that will display the information after you install it on your smartphone.