There are, of course, two ways: legal and illegal.

The illegal way is by far the simpler one, and would only require some cash for bribing a morally reprehensible captain, who may in turn suborn a number of civil servants from the Casa de Contratación de las Indias so they do not thoroughly inspect his ship. With that, you would be set to go.

The legal process is more complex, of course, because bureaucracy is what it is. For going to the Indies a license was required. This license was issued by the Casa de Contratación de las Indias, in Seville, after a certain process through which they would examine the person's "quality", which is to mean that the traveller is of clean blood, doesn't have a criminal record, does not have any debts to the Crown or does not have any debts of judiciary nature. The law was quite explicit on the matter:

We command that no "reconciliated", nor son of a man who had publicly worn a "sambenito", nor son or grandson of a man burnt or sentenced for heretic pravity or apostasy on the male or female line, may pass or shall pass to our Indies or its adjacent isles, under penalty of confiscation of all his goods for our chamber and treasury, and loss of our mercy, and be proscribed from the Indies in perpeuity, and if he lacked any goods, shall he receive one hundred whiplashes.

There two groups exempt from the "proofs of blood cleanliness" because it was assumed they were of clean blood and that they would have accredited it before the administrations on earlier dates: ecclesiastic people, and civil servants.

So, if you have the appropriate paperwork and present it to the Casa de la Contratación de las Indias, they may grant you a license, which would be easier if you greased up the bureaucratic machinery, so to speak. With that license in hand, you can try finding a captain that would go to the Indies and be done with it. Once you have foud your captain, you register in his ship as passenger, you are done. It is then up to the captain to hand a copy of the ship's manifest (passengers, crew, and cargo) to the officers of the Casa de la Contratación, but that is none of your concer, though. In the case of a lack of the required paperwork, you can always trust the flourishing criminal world of Seville, where forgers may be willing to craft the documents you need, for a fee, of course.

This, of course, would be the case if you were a single man. If you were married and were willing to settle in the Indies, you were mandated to bring your family with you. This was put in place ir order to avoid abandonment and/or bigamy.

There would also be a third way: enlisting as a soldier on a regiment or company destined to the Indies, or enrolling as a sailor on a ship going there. You can imagine where this is going: when you get to an American port and you leave the ship, you just go missing, moving somewhere else, never to be found by your captain. If he ever finds you, you will be trated as a deserter and hanged.