How to take advantage of the free transaction relay policy?
By simply adding addnode=68.168.105.168 to your bitcoin.conf file then any transaction you send will automatically be broadcast and eventually make it into a block. No one but you has to add this node and nothing special needs to be done beyond this one configuration.

Now it may be true that the bitcoin-qt client will not actually allow you to send a transaction that is against its own rules, I'm not sure as I've never tried, but if this is the case you will need a non-standard client which is another question entirely.

Note: May 2015, 68.168.105.168 is the only Eligius relay IP I can find that's active