Install requirements

GNU C++ Compiler (g++)
libSDL runtime & development files
NASM (Assembler)
GTK+ 2.0 runtime & development files
libVorbis development files
—
Subversion (SVN) – to download the latest EDuke32 source
Timidity – software MIDI player

First check you have the build tools installed by running

gcc –version

You should be given the version, if you get something like ‘command not found’ you will need to install the build tools.
If the version is below 4.0 you should upgrade your build tools to version 4.3 or greater.

If you using Ubuntu you can use the following command to install the build tools.

sudo apt-get install build-essential

If you not using Ubuntu you should search your distributions package manager for the build tools.

All the required development files can be installed on Ubuntu using the following, again check your package manager if you are using a different distribution.

sudo apt-get install subversion timidity libsdl1.2-dev libsdl-image1.2-dev libsdl-mixer1.2-dev nasm libgtk2.0-dev libvorbis-dev

Getting EDuke32

Now you need to get the latest version of EDuke32.

cd ~
svn co https://eduke32.svn.sourceforge.net/svnroot/eduke32 eduke32

I assume your going to build in your home directory, if not I suggest you use /usr/src

Building EDuke32

Once EDuke32 is downloaded you can build it.

cd eduke32/polymer/eduke32
make

If the build completes without errors you should find two executables in your current directory.
Next you need to make the directory where you will place EDuke32 and Duke Nukem 3D.

mkdir ~/duke3d
mv eduke32 ~/duke3d
mv mapster32 ~/duke3d
cd ~/duke3d

Using EDuke32

Alternatively you may install EDuke32 globally, (I.E in /usr/bin) for information on doing this read the EDuke32 wiki.
Now simply get your version of Duke Nukem 3D and look for the file called DUKE3D.GRP. Move this file to where the EDuke32 executable is located. (I.E ~/duke3d)
Once that is done simply run EDuke32 with:

./eduke32

Important Note: ensure that the DUKE3D.GRP file is LOWER CASE, such as duke3d.grp or EDuke32 will not be able to read it, hopefully this will be fixed in future.

Update
There are APT repositories for Debian and Ubuntu for easy installation.

http://wiki.eduke32.com/wiki/APT_repository