I don't know if this is the same man - his CSR which you can order here - would show if it is or not, but it is the only C. K. Atkinson I found:
U.S. Civil War Soldiers, 1861-1865
Name: C.K. Atkinson
Side: Confederate
Regiment State/Origin: Virginia
Regiment Name: 18 Virginia Infantry
Regiment Name Expanded: 18th Regiment, Virginia Infantry
COMPANY: G
Rank In: Private
Rank In Expanded: Private
Rank Out: Private
Rank Out Expanded: Private
Alternate Name: Charles T./Atkinson
Film Number: M382 roll 2
He's also listed as C.F. Atkinson, same unit. There's also a Chastain C. Atkinson.
18th Regiment, Virginia Infantry
18th Infantry Regiment completed its organization in May, 1861. Its members were recruited at Danville and Farmville, and in the counties of Nottoway, Cumberland, Prince Edward, Appomattox, Pittsylvania, and Charlotte. The unit fought at First Manassas under General Cocke, then was assigned to General Pickett's, Garnett's, and Hunton's Brigade. It participated in the campaigns of the Army of Northern Virginia from Williamsburg to Gettysburg except when it was detached to Suffolk with Longstreet. Later it served in North Carolina, returned to Virginia, and took an active part in the battles of Drewry's Bluff and Cold Harbor. The 18th endured the hardships of the Petersburg trenches north of the James River and saw action around Appomattox. It lost 6 killed and 13 wounded at First Manassas and in April, 1862, had 700 men fit for duty. The unit reported 206 casualties during the Seven Days' Battles, and of the 120 engaged in the Maryland Campaign, thirty-six percent of the 312 in action were killed, wounded, or missing. Many were captured at Sayler's Creek and only 2 officers and 32 men surrendered. The field officers were Colonels Henry A. Carrington and Robert E. Withers, Lieutenant Colonel George C. Cabell, and Major Edwin G. Wall.
Name: Chastain Atkinson
Enlistment Date: 28 Apr 1864
Enlistment Place: Nottoway County, Virginia
Side Served: Confederacy
State Served: Virginia
Service Record: Enlisted as a Private on 28 April 1864.
Enlisted in Company G, 18th Infantry Regiment Virginia on 28 Apr 1864.
Name: C.K. Atkinson
Side: Confederate
Regiment State/Origin: Virginia
Regiment Name: 18 Virginia Infantry
Regiment Name Expanded: 18th Regiment, Virginia Infantry
COMPANY: G
Rank In: Private
Rank In Expanded: Private
Rank Out: Private
Rank Out Expanded: Private
Alternate Name: Charles T./Atkinson
Film Number: M382 roll 2
Name: Charles Atkinson
Occupation: Carpenter
Enlistment Date: 22 Apr 1861
Enlistment Place: Nottoway County, Virginia
Side Served: Confederacy
State Served: Virginia
Service Record: Enlisted as a Private on 22 April 1861 at the age of 28. [b. abt 1833]
Enlisted in Company G, 18th Infantry Regiment Virginia on 22 Apr 1861.
JR