Here is a simple solution to Blender missing 'libavcodec.so.54' shared library on Ubuntu Xenial (16.04.1 LTS) box. There are numerous posts about this problem but very few solutions.
$ blender
blender: error while loading shared libraries: libavcodec.so.54: cannot open shared object file: No such file or directory
Solution for Blender missing library
Let me share what worked for me. Let's try to locate and add the library to ldconf path.
# Install the libavcodec-extra package using apt-get package manager.
$ sudo apt-get install libavcodec-extra-56
# Verify the location of libavcodec.so.54 file path.
$locate libavcodec.so
/usr/lib/x86_64-linux-gnu/libavcodec.so
/usr/lib/x86_64-linux-gnu/libavcodec.so.56
/usr/lib/x86_64-linux-gnu/libavcodec.so.56.1.0
# Create a new ldconfig file with proper path to the shared library.
$ sudo nano /etc/ld.so.conf.d/blender.conf
# Add the following line in the file.
/usr/lib/x86_64-linux-gnu/
# Rerun ldconf
$ sudo ldconf
# Start blender normally.
$ blender
That's all folks!. Enjoy blender.