The murderer is the man (one) who was driving (drove) the car that day.
你这句话2个错误,
1、that是指示代词,不做宾语用.宾语是那个人,可以用the man（名词作宾语）或者the one（代词作宾语）.
2、who带的是定语从句,这个从句应该是以who为主语的一个完整句子.你的原句：who driving the car不是完整句子,因为driving是动名词,不可以做谓语.因此,改为was driving过去进行时或者用drove过去式当谓语.那么who为主语,was driving或drove为谓语,the car为宾语.