There are a number of possibilities to obtain European citizenship.
One of them is citizenship by descent, known as jus sanguinis.
If you are able to demonstrate that you have ancestors from Europe, in many cases you can be entitled to citizenship. This wont cost you much but it can take a lot of time – anything between 6 months to 2 years of paperwork.
If this is not an option, another way is to establish legal residence in one of the European countries and wait.
Depending on the country it can take between 5 and 10 years until you can apply for citizenship. This period can be reduced in some cases depending on your specific situation
Obtaining residency may prove to be the real challenge.
Marrying an EU citizen or finding a job in Europe will entitle you to set residence in Europe, another option would be to make a substantial financial investment, to create jobs or in some cases to buy property in Europe.
Portugal for instances, by means of its Golden Visa Scheme, will give you legal residency when you buy property for €500.000 or more.