I will be shortly pursuing a degree course in computing and I would like to obtain first the entry qualification that is required for the degree course.

pursuing 是动词 pursue 的现在进行时形式。pursue 由前缀 pur-（表示"向前"）和 sue（来自拉丁语 sequi，意为"跟随"）组成，本义是"追随、追求"，引申为"从事、进行（学业或事业）"。 在这里 pursuing a degree course 表示"攻读学位课程"的意思。