They're not actually referred to by the same name. The 'Han' in the Han Dynasty and the 'Han' in Han2guo2 are actually different characters that are also pronounced differently.

As is pretty clear from comparing Korean cultural practices and language with Han Chinese, the influence of Han Chinese culture is definitely one of the foundations of Korean identity throughout history. The almost prehistoric Joseon (~300 to 100 BCE) and later Goguryeo (~37 BCE to 668 AD) states aren't any different.

While the early history of Joseon is extremely shaky and surrounded by myths and legends, the most reasonable (in my opinion) is that the state of Joseon was founded by Weiman, a military official from Yan during the Zhou dynasty, who fled to what is now the northern Korean peninsula and allegedly founded a settlement centered at what is now modern-day Pyongyang.

Why is this the most reasonable? Well, the first indisputable evidence of Joseon comes when Han China invades the Korean peninsula after a conflict with the alleged grandson of Weiman. That, and because the two other major theories involve people turning into bears. As such, the Weiman theory is generally accepted amongst historians (Barnes 2001).

Nevertheless, it is indisputable that China had a major part in Korea's foundation myths and, as evident in Chinese influences on the Korean language and cultural practices, in what it is today.

I took most of my sources from Gina Lee Barnes' book "State Formation in Korea: Historical and Archaeological Perspectives", published in 2001. The first few pages are available on Google Books and give some high-calorie breakdowns of the foundation of pretty much every Korean state during the early historic periods.