那天开车的人就是凶手.the murderer is that who driving the car that day

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.