How do I get apt-get to ignore some dependencies?

Simple and easy solution: just specify the unwanted packages with an extra - after each of them.

Example without the - switch:

root@debian:~# apt-get install bsd-mailx 
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following extra packages will be installed:
  exim4-base exim4-config exim4-daemon-light liblockfile-bin liblockfile1
[...]

Example using the switch to avoid installing exim4-base. Notice the - at the end:

root@debian:~# apt-get install bsd-mailx exim4-base-
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Package 'exim4-base' is not installed, so not removed
The following extra packages will be installed:
  liblockfile-bin liblockfile1 ssmtp
[...]

As you can see, apt-get does not try anymore to install the exim4-base package, and it does not try to install its various dependencies (exim4-config etc).

And if you were wrong and needed that exim4-base dependency after all, you can just apt-get install it later!

You can change the dependencies of a deb package like this:

1. Unpack deb: ar x golden-linux.deb (will create i.e. three files: debian-binary control.tar.gz data.tar.gz)
2. Unpack control archive: tar xzf control.tar.gz (will create: postinst postrm preinst prerm md5sums control)
3. Fix dependencies in control (use a text editor)
4. Repack control.tar.gz: tar --ignore-failed-read -cvzf control.tar.gz {post,pre}{inst,rm} md5sums control
5. Repack deb: ar rcs newpackage.deb debian-binary control.tar.gz data.tar.gz (order important! See [Note] )

[Note]: dpkg wouldn't be able to read the metadata of a package quickly if it had to search for where the data section ended!

After you install the package with the --ignore-depends option, go and edit the /var/lib/dpkg/status file, and remove whatever dependency you think is not needed. Just be very careful. In order a dep. to be required, it is more than likely to BE required

You can try the --nodeps flag with apt-get.
Or download the package and install it using dpkg with the option --ignore-depends.

For example, if you want to install package foo without dependency bar:

dpkg --ignore-depends=bar -i foo_1.2.3_amd64.deb

Since you installed postfix from source, you need to install a "dummy" package which will satisfy the mail-transport-agent dependency of mailx (or bsd-mailx). The "equivs" package in debian exists to create such a dummy package which you can install to tell dpkg "this dependency is satisfied"

The reason that telling dpkg to simply ignore dependencies is not a good solution, is that you are only telling dpkg/apt to ignore it for a single transaction, you can't tell it to ignore dependencies forever. Everytime you use apt it checks the dependencies on all packages