How much did class conflict cause the English Civil War?

Hardly at all, despite the efforts of some later commentators to put a Marxist "spin" on it. Many of what would be regarded as the "lower classes" felt a deep antipathy towards opposing their own King, and in contrast (on the Parliamentarian side) the likes of Lord Manchester, Sir Thomas Fairfax, the Earl of Essex and even Cromwell and his son-in-law Henry Ireton (land-owner, MP and lawyer)were hardly of the "riff-raff"!

[HK: to categorize any class as having a united view would of course be incorrect. The mistaken belief that middle trading classes were regarded as anything but 'riff-raff' by the king or even lords would also be an error despite what some of those middle classes may delude themselves of even to this day.]

The fundamental causes of the conflict were (a) money, and (b) religion.

King Charles's basic problem as regards money was that he didn't have any, and could only attempt to raise some by way of taxes which Parliament refused to agree to except on a short-term basis.

There was probably a religious aspect even to this, as it was a time of strong anti-Catholic feeling in England and Scotland, and many were deeply suspicious of the fact of the king having a Catholic Queen (albeit by an arranged marriage). When Charles enthroned Archbishop Laud at Canterbury, and the church began to re-instate some of the finery within the Church and its ceremonies which was widely regarded as Popish idolatry, there was even a widespread view that Charles himself was a closet Catholic.

Against this growing suspicion and hostility there was the (ultimately fatal) contrast of the King's own stubborn belief in the "Divine Right of Kings". His view that he was annointed by God to rule without question, without reference to or approval by Parliament or anyone else led inevitably to the trial of strength which became the Civil War.

Even at a remarkably late stage in the conflict he could probably have kept both his head and even his crown if had had the least concept of the art of compromise, but such was not to be, and the foundation of the much later concept of constitutional monarchy was founded.