According to the Roman Catholic church what happens to a non-believer when they die?
I believe Rome teaches that a person must have partaken of the sacraments, and believe in Christ.
If any one saith that the sacraments of the New Law are not necessary unto salvation...and that without them, or without the desire thereof, men obtain from God, through faith alone, the grace of justification...let him be anathema (The Canons and Decrees of the Council of Trent. Found in Philip Schaff, The Creeds of Christendom (Grand Rapids: Baker, 1919), Canon IV, p. 119).
In addition to this penance would also be necessary:
Penance is...necessary because we must expiate and make reparation for the punishment which is due our sins...We make satisfaction for our sins by every good act we perform in the state of grace but especially by prayer, penance and the practice of charity (John Hardon, The Question and Answer Catholic Catechism (Garden City: Image, 1981), Question #1320).
The mass is also necessary to cleanse from sins committed after baptism:
In this divine sacrifice...that same Christ is contained and immolated in an unbloody manner who once offered himself in a bloody manner on the altar of the cross...This sacrifice is truly propitiatory...If any one saith, that the sacrifice of the mass is only a sacrifice of praise and thanksgiving; or that it is a bare commemoration of the sacrifice consummated on the cross, but not a propitiatory sacrifice...and that it ought not to be offered for the living and dead for sins, pains, satisfactions and other necessities: let him be anathema (The Canons and Decrees of the Council of Trent. Found in Philip Schaff, The Creeds of Christendom (Grand Rapids: Baker, 1910), Doctrine on the Sacrifice of the Mass, Chp. II, p. 180, Canon III).
So, if an unbeliever did not do these things he would be hell bound according to Rome.
You also asked:
Is there any difference for a non-believer who has been baptised compared to a non-believer who hasn't been baptised?
One must first ask the question, 'Is Catholic baptism valid?'. I do not believe it is. But even assuming that it was, according to the Catholic position, the person would still be bound to hell, as he has not partaken of baptism and has no faith in Christ.
Now, for the sake of openness, I will state that I am not a Catholic, but a reformed protestant.