How to get Android crash logs?

You can try this from the console:

adb logcat --buffer=crash 

More info on this option:

adb logcat --help

...

  -b <buffer>, --buffer=<buffer>         Request alternate ring buffer, 'main',
                  'system', 'radio', 'events', 'crash', 'default' or 'all'.
                  Multiple -b parameters or comma separated list of buffers are
                  allowed. Buffers interleaved. Default -b main,system,crash.

If your app is being downloaded by other people and crashing on remote devices, you may want to look into an Android error reporting library (referenced in this SO post). If it's just on your own local device, you can use LogCat. Even if the device wasn't connected to a host machine when the crash occurred, connecting the device and issuing an adb logcat command will download the entire logcat history (at least to the extent that it is buffered which is usually a loooot of log data, it's just not infinite). Do either of those options answer your question? If not can you attempt to clarify what you're looking for a bit more?