I know this post is a bit old, but I found an answer here for anybody that may stumble upon this post: https://help.ubuntu.com/community/RemoveOldKernels

In case that link is broken, here is the relevant snippet:

Safely Removing Old Kernels

For users of LVM systems, encrypted systems or limited-storage systems, the most frequent problem is that the /boot partition is simply full. The package manager cannot install a pending upgrade due to lack of space. Besides, apt-get can not remove a package due to broken dependency.

This problem can be fixed quickly and easily from the shell. Simply identify one or two old kernels to remove manually, which will provide the package manager enough space to install the queued upgrade.

$ sudo rm -rv ${TMPDIR:-/var/tmp}/mkinitramfs-*  
                                  ## In Ubuntu 16.04 and earlier there may be leftover temporary
                                  ## files to delete.
                                  ## See: https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=814345

$ uname -r                        ## This command identifies the currently-running kernel
4.2.0-21-generic                  ## This is the current kernel.
                                  ## DO NOT REMOVE it!

$ dpkg -l | tail -n +6 | grep -E 'linux-image-[0-9]+' | grep -Fv $(uname -r)
                                  ## This command lists all the kernels excluding the booted
                                  ## kernel in the package database, and their status.
rc  linux-image-4.2.0-14-generic  ## The oldest kernel in the database
                                  ## Status 'rc' means it's already been removed
ii  linux-image-4.2.0-15-generic  ## The oldest installed kernel. Eligible for removal.
                                  ## Status 'ii' means Installed.
ii  linux-image-4.2.0-16-generic  ## Another old installed kernel. Eligible for removal
ii  linux-image-4.2.0-18-generic  ## Another old installed kernel. Eligible for removal
ii  linux-image-4.2.0-19-generic  ## The previous good kernel. Keep
iU  linux-image-4.2.0-22-generic  ## DO NOT REMOVE. Status 'iU' means it's not installed,
                                  ## but queued for install in apt.
                                  ## This is the package we want apt to install.

                                  ## Purge the oldest kernel package using dpkg instead of apt.
                                  ## First you need to remove the image initrd.img file manually
                                  ## due to Bug #1678187.
$ sudo update-initramfs -d -k 4.2.0-15-generic
$ sudo dpkg --purge linux-image-4.2.0-15-generic linux-image-extra-4.2.0-15-generic
                                  ## If the previous command fails, some installed package
                                  ## depends on the kernel. The output of dpkg tells the name
                                  ## of the package. Purge it first.

                                  ## Also purge the respective header package.
$ sudo dpkg --purge linux-headers-4.2.0-15-generic
                                  ## Try also purging the common header package.
$ sudo dpkg --purge linux-headers-4.2.0-15
                                  ## Do not worry, if the previous command fails.

$ sudo apt-get -f install         ## Try to fix the broken dependency.

I followed this with:

sudo apt-get autoremove --purge

As always in Linux, there's more than one way to get the job done. However, if you need to do it from CLI, this is my preferred method:

I start by running this as root or with sudo:

du -cha --max-depth=1 / | grep -E "M|G"

The grep is to limit the returning lines to those which return with values in the Megabyte or Gigabyte range. If your disks are big enough, you could add |T as well to include Terabyte amounts. You may get some errors on /proc, /sys, and/or /dev since they are not real files on disk. However, it should still provide valid output for the rest of the directories in root. After you find the biggest ones you can then run the command inside of that directory in order to narrow your way down the culprit. So for example, if /var was the biggest you could do it like this next:

du -cha --max-depth=1 /var | grep -E "M|G"

That should lead you to the problem children!

Additional Considerations

While the above command will certainly do the trick, I had some constructive criticism in the comments below that pointed out some things you could also include.

1. The grep I provided could result in the occasional "K" value being returned if the name of the directory or file has a capital G or M. If you absolutely don't want any of the K valued directories showing up you'd want to up your regex game to be more creative and complex. e.g. grep -E "^[0-9\.]*[MG]"

2. If you know which drive is the issue and it has other mounted drives on top of it that you don't want to waste time including in your search, you could add the -x flag to your du command. Man page description of that flag:

     -x, --one-file-system
         skip directories on different file systems
   

3. You can sort the output of the du command so that the highest value is at the bottom. Just append this to the end of the command: | sort -h