If the 1850 census is correct, Catherine BEAN is William's second wife and not the mother of his children.

1.  Marriage Record:

2.  1820

3.  1830 Census Index/Images (online at Ancestry.com):

4.  1840 Census Index/Images (online at Ancestry.com):  there's a William KLINGER indexed in Shuylkill Co., PA, and a William CLINGER in Crawford Co., OH, but none in Preble Co., OH.  I did a page-by-page check of Lanier Twp., Preble Co., OH, and found a Nicholas KLINGER (mis-indexed KILINGER) and a Michael KLINGER, who are William's brothers.

5.  1850 Census Index/Images (online at Ancestry.com):  can't find, and I did a page-by-page check of Lanier Twp., Preble Co., OH, where I do find William's brother, Michael KLINGER, living near Andrew & Matilda (KLINGER) COPP and PAULLUS and LOY.  I also did a page-by-page of Twin Twp., where I found a young William KLINGER (æ 26, b. OH), who was not indexed, living near our subject's son, Isaac KLINGER.  I also found:

.  1850 Census Index/Images (online at Ancestry.com; Image #38 of 47):  West Alexander, Twin Twp., Preble Co., OH, p. 442A, 848/848, enumerated 5 Sep 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):

William Klinger	60	M		None	8300	Pennsylvania
Caroline Klinger	56	F				"
Margaret Klinger	16	F				Ohio

William's wife should be Catharine, born in Virginia.  Has this enumerator made multiple errors?  Or did William remarry between 1850 and 1860?  Living four households from Adam LOY and his wife, Elizabeth (æ 50, b. VA), with their son, Alfred (æ 15).

6.  1860 Census Index/Images (online at Genealogy.com; Image #6-7 of 48):  Eaton P.O., Twin Twp., Preble Co., OH, Roll 1026 (Book 1), pp. 3B-4A, PN 6-7, 45/45, enumerated 2 Jun 1860, official enumeration date 1 Jun 1860 (extracted by Diana Gale Matthiesen):

William Klinger	66	M		Gentleman	730	2320	Pennsylvania
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Catharine Klinger	68	F					Virginia

Living next door to Adam LOY and his wife, Elizabeth (æ 59, b. VA), with their son, Alfred (æ 24).  In the 1850 census, there is an Alfred LOY (æ 12), living with our subject's son, Isaac KLINGER, and his wife.  I'll wager that Mrs. Elizabeth LOY is Catharine's sister.

7.  1870

8.  1880 Census Images (online at Ancestry.com, Image #2 of 43):  Lanier Twp., Preble Co., OH, Roll 1060, p. 133B, PN 2, SD 2, ED 200, (this enumerated used no visitation numbers, for anyone), enumerated 1 Jun 1880, official enumeration date 1 Jun 1880 (extracted by Diana Gale Matthiesen):¤•

1880:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).
3	4	5	6	8	9	10	11	13	24	25	26
Andrew Copp	W	M	57			/		Farmer	Virginia	Virginia	Virginia
Matilda E "	W	F	55	Wife		/		House Keeper	Ohio	Pennsylvania	Virginia
John Hoops	W	M	22		/			Farmer	Ohio	Hanover	(Hanover)
Catharine Klinger	W	F	89	mother law			/		Virginia	(Virginia)	(Pennsylvania)

9.  Archives of the PANORTHU-L mailing list (online at RootsWeb.com), specifically a post by Jo Kester, which included:

1   Philip KLINGER b: Nov 10, 1764 GER  d Apr 10, 1844 ..  +Barbara ECOS/EKER b Jun 26, 1759  d Oct 02, 1846 . 2   William KLINGER b: April 12, 1793  d: January 02, 1864 .....  +Catherine BEAN b: 1791 VA  d: 1881 .... 3   Isaac KLINGER b: March 06, 1825  Preble, OH            d: April 07, 1873 Palestine, Darke, OH ........  +Elizabeth COPP b Apr 22, 1826 Shendoah,VA              m: Aug 27, 1846 Preble Co, OH              d Jul 07, 1892 Palestine, Darke, OH ........ 4   Mary Catherine "Kate" KLINGER               b Mar 26, 1847 West Alexandria, Pre, OH                   d Jul 05, 1939 Greenville, Darke, OH         ..  +Charles Washington KESTER                 b Mar 28, 1847 Palestine, Dar, OH                 m Dec 26, 1867 Eaton, Pre, OH                 d Feb 17, 1923 Union City Wash Twp, Dar, OH