A good place to start is with something such as an MTR or ping when the issue is occurring. This will give you an idea if the droplet is responding to anything at all.
For logs, it's hard to say, as you will want to get an idea of what's going on first. For the most part, most logs are found in the
/var/log/ folder. You can find a great guide on viewing logs here
If I had to guess, something is using up all your resources. It seems like it based on the symptoms. So You may want to check the logs of services running. The
dmesg command may also be of help, as it may indicate errors with memory, CPU, disk, etc that may shed some light.
Linux and the applications that run on it can generate all different types of messages, which are recorded in various log files. Linux uses a set of configuration files, directories, programs, commands and daemons to create, store and recycle these log messages. Knowing where the system keeps its log files and how to make use of related commands can therefore help save valuable time during troubleshooting.