Kay,
Here is a basic description of the duties of a battery commander.

CAPTAIN (Battery Commander)
Had overall command, control, and responsibility for the training, serviceability, and combat operation of the battery's personnel and equipment. He was not only the chief recruiter of the company, but also used his influence to acquire horses and other material, through means outside normal requisitions, to keep his battery in the best possible condition. Depending on the organization of the army at a particular time, the captain received his orders from either an artillery battalion commander, a division "Chief of Artillery", or an infantry brigade commander; ranking from major to brigadier general respectively. The captain had command over as many as 170 men and 98 horses in a six gun battery with six horse teams. In a four gun battery with four horse teams he had to have a minimum of 71 men and 45 horses to function efficiently. Most artillery officers were very slow to receive promotion due to the relatively light casualties and "turnover" in the long arm as compared to the infantry.

As to the qualities that a Private would need to be promoted to Captian.A lot of variables could enter this selection, among which could be the following;

1) Attrition of others in the Battery due to death/injury/disease.There generally wasn't a steady supply of new replacements comming into Civil War companies-so the available men to fill positions were limited to those already in the organization.

2) Political affiliations and social connections.Who a person knew politically, or where they stood in the local social standing often eased their commission as an officer.

3) Exemptalary preformance of duties, to such extent that he recieved the notice of superior officers.The qualities exhibited would be the same as for anyone who recieves promotion in any organization, competence, organizational skills,ability, etc.

Or a combination of these and other variables.