DISCLAIMER: I’m not a native speaker, so please bear with my English even if it may sound clumsy.
The latin christendom of the mid to late 11th and early 12th century was largely defined by the Gregorian Reform, named after Pope Gregory VII (the movement started before Gregory’s papacy, though). Rooted in western France, namely in the famous abbey of Cluny, the ideas of the reform spread in western Europe, especially in France and the (Holy) Roman Empire. Members of the movement saw a decline in piety of the clerics and criticized practices like simony – i.e. receiving a clerical office for money – or the marriage of priests.
In his reign as king of England (1066–1087) William transformed the church heavily in three aspects:
At the end of his reign, out of fifteen English bishoprics eleven were in the hand of Normans. Most of these got vacant due to old age, but in some cases the king stripped the bishops from their office. The most famous case would be the archbishop of Canterbury, Stigand, who supposedly crowned Harold of Wessex. He was deposed by an English council in 1070, which was chaired by a papal legate. He held the archbishopric of Canterbury and at the same time the bishopric of Winchester, which in the eyes of the reform movement was a serious offence. William filled the vacant see with the Lombard Lanfranc of Bec, who was formerly the abbot of Caen and a leading intellectual of his time.
William used the bishoprics as a tool of patronage. He appointed men whom he could trust and who already proofed themselves in his services. Ten of the invested bishops were former royal clerks. A certain Walkelin, former chaplain, became bishop of Winchester, one Herfast, bishop of Thetford, used to be the royal chancellor. According to Frank Barlow, Loyalty in his subordinates was the preferred characteristic of the Conqueror (reminds you of someone?). This helped William to tighten his grip on the kingdom which was by no means entirely conquered after Hastings. Douglas however points out that most of the men William appointed were more than able clerics.
On a more rural level however there wasn’t that much change. Priests remained largely Anglo-Saxon during William’s time. If anything, there was now a gap between the clerical and now Norman elite and the mainly Anglo-Saxon staff of the church.
Many of the now Norman bishops moved their sees from rural areas inside major cities, for example Elmham to Thetford, Sherborne to Salisbury, Selsey to Chichester, Lichfield to Chester or Dorchester to Lincoln. Some bishops already did that before 1066, but a council in 1070 enacted that every bishop must have a permanent see instead of journeying through his bishopric and the Norman bishops moved into the major cities like they were used to from the mainland. This was favourable for William, because the bishops owed military service and it was much easier and quicker to raise troops in a major city. This may be seen as a very early step to a centralised government.
Another structural change was the dispute between the archbishops of Canterbury and York – Lanfranc and Thomas – about the primacy in the English Church. In the end this was decided by William in favour of Lanfranc and in this position he was a great help to his king. There were also administrative changes. A council in 1070 decided that every bishop had to establish the position of an archdeacon. The archdeacons were already a thing in Normandy, even though it is not clear what their duties were. In England however they tended to the administrative tasks in the bishoprics and in this they were so successful that in the mid 12th century chroniclers complained about them raising too much money from the people.
Before William became king of England he already encouraged councils and synods in the duchy of Normandy. In England this practice was enforced from 1070 onwards. This was in accordance with the reform movement and happened seven times during William’s reign. The king was presiding the councils, in England and Normandy alike, and he knew how to use this influence. A council in Winchester 1072 decided that no one should betray king or kingdom and that traitors were to be excommunicated by the bishops in their bishoprics.
King and bishops shouldn’t be seen as entirely opportunistic, though. The councils enacted a lot of rulings in accordance with the reform, moved against simony and, by enforcing the celibate, against nicolaitism.
During his reign William released a writ in which he declared that church courts no longer were to pass legal cases in their jurisdiction to the hundred courts. This was by no means a “seperation of church and state” as it was regarding the hundred courts only and not the shire courts. In the eyes of the contemporaries it was a strengthening of the bishoprics into what was believed to be a former state of their legal rights.
Long story short: Together with Lanfranc William promoted a centralisation of the English church and bishoprics by “normanising” the church, enforcing cleric jurisdictions and incorporating leading clerics into the government.
Some further Reading:
BARLOW, Frank, The English Church. 1066-1154, New York 1979.
BARTLETT, Robert, England under the Norman and Angevin Kings. 1075-1225, Oxford 2000, esp. pp. 377–441.
HARPER-BILL, Christopher, The Anglo-Norman Church, in: A Companion to the Anglo-Norman World, ed. Christopher HARPER-BILL and Elisabeth VAN HOUTS, Woodbridge 2003, pp. 165–190.
HUSCROFT, Richard, The Norman Conquest. A New Introduction, Harlow u.a. 2009, esp. pp. 289–313.
You might also look into the recently published BATES, David, William the Conqueror, Yale 2016. I hadn’t have a chance to read it yet, but it should be a good read.