On my Windows machine and Ubuntu laptop I use KeePass and KeePassX for working with my KeePass (.kdbx). I keep the database in the cloud for convenience. When I got my Chromebook I first started using a Chrome browser extension called CKP – KeePass integration for Chrome. It is very convenient in the way that it always sits in you Chrome toolbar and suggests you which entry you are looking for depending on the tab you have currently open (it does a pretty good job) and then offers to autofill in the credentials. This is all fine, but it lack the abilty to add an entry to the database. Again, there is a solution in the Android Play store for this, if you own a ChromeOS device that can run Android apps.
If you have an Android capable chrome device – mine is an Acer Chromebook R11, but all 2017 chrome devices are Android capable – open the Google Play Store and install KeePassDroid. If you have your kdbx file in the cloud you will need to install the app for this respective service as well. So for Google Drive for instance install the Google Drive Android app. Now, if you want to be able to write to the kdbx file it is important (see this bug report for details) that you open the kdbx file via the Android Google Drive app, it will ask you to open it with KeePassDroid, confirm and type you password.
That’s all, you are ready to go.