Make a name for yourself. Royalty is a general term which is used differently in different contexts. Everything from the world of hip-hop to your local country club designates some family or group of people as the royalty. Find a niche for yourself where you can be considered royalty to make a first and decisive impression on other royals.
Affect an air of indifference and even contempt. The only thing more off-putting to a member of a royal family than a commoner is someone who acts commonly. Even if you're just an everyday Joe, act as if you're the King or Queen of Siam. Royalty has a taste for other royalty, so play the part to come closer to marrying into a royal family.
Dig up some noble blood lines. Even the thinnest, most brittle connection between you and some royal or noble blood line will get you closer to marrying royalty. Do some genealogy research on your family to find blueness in your blood.
Buy your nobility. If you have enough money, or a desperate enough monarchy, you can buy a noble title or peerage in many places. For the super-wealthy, the United Kingdom is a reasonable market for a duchy or barony. For those less monetarily blessed, think about an Eastern European fiefdom, for example, to find a good noble title. Appearing at a royal dinner or function with a chest full of medals or a noble tiara will greatly increase your chance of marrying royal.
Find an environment where there are royals floating around. Europe, for example, is an excellent place to find the descendants and family members of crumbled royal lineages. These penniless royals are now almost regular people. Find them hanging around cultural centers of Europe and use your best pick-up line and dating strategies to start up with them as you would with any other person. Be yourself, put your best foot forward and you might find yourself marrying into a royal family.