I'll try and keep it brief, but to understand more completely it's necessary to talk about the economic state of the United States during this time period.

Also, there are three main ways a government may fund a war: taxation, borrowing, or printing more money. They are not exclusive and governments may do any combination thereof. The Union government relied primarily on borrowing (in the form of bonds) and taxation (through tariffs), but they also printed "Greenbacks" to stimulate the economy.

Pre-War:

Northern Economy - Influx of immigrants from Europe allowed Northern states the luxury of cheap, willing, and able workers. Combine that with an economy largely centered around industrial factories (industrialization will be gearing up all the way into the 20th Century) and you end up with an economy that produces a lot of finished products from raw material. Finished products would then be distributed and sold in mainly domestic markets.

Southern states - Economy largely based on cash crops grown at plantations such as cotton, tobacco, and sugar cane. The institution of slavery allowed plantation owners huge economic benefit over poor-whites who practiced sustenance farming. Plantation owners would keep what they needed, but try to sell most of cash crop to European countries.

This difference in economic models eventually led to a disagreement on how the federal government should become funded. Industrial states in the North were in support of higher tariffs because they could produce anything they wanted while Southern states favored lower tariffs because they were forced to import many goods.

War:

With the Southern states and their low tariff desires out of the picture, the North was able to pass pro-tariff legislation (do a search for the Morrill Tariff Act) and raise taxes and tariffs to unprecedented levels. Income began to be taxed during this time as well as legal documents (think back to the Stamp Taxation during Colonial times) and liquor. According to the link supra, 21% of the Union's revenue came through taxation.

According to this source, nearly $3 Billion dollars of revenue was generated by the Union government through the sale of War Bonds. These bonds primarily targeted banks and wealthy citizens, but also targeted the middle class through the use of patriotic propaganda.

The use of War Bonds in this example is interesting for two main reasons: 1) It proved to be effective and thus served as precedent for how many governments, including the US government, would raise finances during times of war; and 2) the introduction of the Legal Tender Act allowed for the use of paper money NOT backed up by silver or gold.

tl;dr: As with any country funding a war, the US funded the Civil War through a combination of taxation, borrowing, and printing money. Taxation accounted for about 1/5 of the Federal government's revenue during this time, War Bonds accounted for about 2/3 of the Federal government's revenue, and the remainder came through the printing of paper money through the Legal Tender Act of 1862