Download Chrome for Mac. For Mac OS X 10.10 or later. This computer will no longer receive Google Chrome updates because Mac OS X 10.6 - 10.9 are no longer supported.
You need some more RAM, but:
Start by checking if you can run Snow Leopard:
Requirements for OS X 10.6 'Snow Leopard'
OS 10.6 Snow Leopard is once again available from the Apple Store:
Download Mac Os X 10 11 4
and in the UK:
but nobody knows for how long it will be available.
To use iCloud you have to upgrade all the way to Mountain Lion:
You can also purchase the code to use to download Lion (Lion requires an Intel-based Mac with a Core 2 Duo, i3, i5, i7 or Xeon processor and 2GB of RAM, running the latest version of Snow Leopard), or you can purchase Mountain Lion from the App Store - if you can run that:
Google Chrome For Mac Os X 10.4 Tiger Download
Download Mac Os X 10.4 Tiger
When you have installed it, run Software Update to download and install the latest updates for Snow Leopard.