Honestly, this is one of the easiest COD games to beat veteran in. I recommend 2 play throughs. Once on recruit to get the intel, mission specific achievements, etc. And the second play through to get veteran, and any achievements you missed (such as fully upgrading the Exo suit).
The reason I say two play throughs is because by the time you beat the campaign on recruit you will A) know what to do and where to go and B) you will have your exo suit nearly maxed out. Just go to mission select, choose the first mission on veteran and begin.
I recommend (on your recruit play through) to get fast reload, more health, and more grenades upgraded as soon as possible, as they will make it so much easier to get it done.