How and when did the Republican Party begin?

The Republican Party was founded in 1854. It was an offshoot of the Whig Party, a group that espoused classic liberalism, a set of beliefs that emphasized the values of a free market economy and personal liberty in matters of religion and other forms of civic life. One important issue for the Whigs is that they had their origins in England, where there was a state church, the Church of England. It was not until 1854 that Dissenters (people not members of the Church of England) could attend Oxford University. Dissenters tended to support the Whig Party because of its emphasis on religious freedom. Evangelicals were especially opposed to slavery on religious grounds. Evangelicals also tended to support other radical ideas including temperance (prohibition of alcohol) and women's suffrage.
The key event that led to the formation of the Republican Party was opposition to the Kansas–Nebraska Act of 1854. This bill would have made slavery legal in new territories north of the 36th parallel. The party's main power base was New England and the midwest. As well as being anti-slavery, they were strongly pro-capitalist, arguing for the benefits of system of free labor and wealth earned by industry. The early Republican platform not only opposed slavery in western territories but supported developing banking and railroads and giving free land to pioneering small farmers in the newly acquired western territories. 
The party adopted the name Republican at a convention held on July 6, 1854 in Michigan. It held its first national convention on February 22, 1856 in Pittsburgh and nominated John C. Frémont as its first presidential candidate. 

In 1854, there were several parties (Whigs, Free Democrats, and Independent Democrats) who were opposed to slavery and wanted to abolish it across the country. In June of the same year, Horace Greeley published an editorial which suggested the parties should become united under one name—the Republican Party. He wrote:

...though we think some simple name like 'Republican' would more fitly designate those who had united to restore the Union to its true mission of champion and promulgator of Liberty rather than propagandist of slavery.

The Republican Party grew rapidly, and by 1855 they held the majority of the House. Their first public convention took place in 1856.  
When the Republican Party started, it was the more progressive and liberal of the political parties in the United States. They fought diligently to end slavery and set the stage for Abraham Lincoln to lead the country during the Civil War.
 
https://www.ushistory.org/gop/origins.htm