Since I’m being lazy this post is mostly for me. Here’s a good starting place:
Egghelp.org
After you’ve downloaded the tar.gz file, uploaded it to your server, and pretty much gone to ./configure, then come back here.

If you can go to ./configure and everything runs peachy, you’re a better system setup guy than I am. Otherwise Eggdrop requires TCL to be installed and compiled (you should get some sort of compiler error saying it can’t create executables if not installed).

apt-get install gcc
apt-get install tcl8.4
apt-get install tcl8.4-dev

That’s it. Go and run that ./configure again.

Unless you’re on Ubuntu 14.04…
./configure --with-tclinc=/usr/include/tcl8.6/tcl.h --with-tcllib=/usr/lib/x86_64-linux-gnu/libtcl8.6.so