How to access SysDump menu on Samsung Galaxy S5, Android 5.0
Keep in mind to update apps it needs something like 3x the actual size. Not sure why you want a sysdump/logcat as your only real solution is just to clear space from device. Install an app like DiskUsage that allows you to see how the space is being used and start clearing up space.

As to your actual question if you need a logcat, enable developer options (settings/about and tap 3+ times build number), enable adb and use logcat command through the pc using adb.

Also through adb you can you can run adb shell and dump dmesg (kernel log).