iMessage is designed to work across your Apple devices using your Apple ID, but it does not live inside iCloud as a standalone app you can open separately. You can access iMessage on iCloud in the sense that your conversations sync to iCloud when you enable Messages in iCloud, so your recent messages, groups, and read status appear on every device signed in with the same Apple ID.
This synchronization relies on iCloud infrastructure while requiring each device to run iMessage locally. Below you will find a clear breakdown of how iMessage and iCloud relate, what is and is not stored in iCloud, and practical steps to verify and manage access on Mac, iPhone, iPad, and web.