There are several possibilities of what might comprise the northernmost battle of the Civil War, depending on the scope of the question. Let's have a look at each of these, and then you can choose whichever one you think is appropriate.
The northernmost official land battle that took place during the Civil War was the Battle of Salineville. In June of 1863, Confederate General John Morgan left Tennessee and traveled through Kentucky and Indiana into Ohio with a force of almost 2,500 men. After winning a battle at Corydon in Indiana, Morgan and his men fled into Ohio while being chased by Federal troops. After failing to cross the Ohio River into West Virginia, he was cut off and forced to surrender at Salineville after a last battle.
The northernmost Confederate land action during the Civil War took place at St. Albans, Vermont, on October 19, 1864. It became known as the St. Albans Raid. However, whether it can properly be termed a battle is a matter of opinion. Approximately 25 Confederate soldiers who had been hiding in Canada invaded the town, killed one man, wounded another, robbed three banks, and then escaped back into Canada. Canadian authorities apprehended them but were unable to allow extradition due to Canada's neutrality. They did, however, return as much of the money as they managed to recover.
The Civil War battle that was fought at the northernmost latitude was a naval battle. The battle of Cherbourg took place off the coast of France on June 19, 1864 between the USS Kearsarge and the CSS Alabama, a Confederate warship. After about an hour's battle, the Kearsarge was victorious, as the Alabama began to sink and its crew had to be pulled out of the water by the crew of the Kearsarge and a passing British yacht.