Yes, Patrick was an immigrant to Ireland. A willing immigrant the second time, sent by God and under the Bishop of Rome's authority. It was the Bishop of Rome who gave him his Latin name, Patricius, meaning "patrician," a Roman nobleman. His birth name was something like Maewin or Succat (accounts differ). He was born (probably) in Cumbria in the last years (probably) of the Fourth Century. His father was a local Briton who held the Roman citizenship, a member of the decurial class who was also a deacon in the Church.
Patrick would have thought of himself as a Briton, but also as a Roman. The Empire was still going in the West at that time. There were four Roman provinces in the lower half of Britannia. Beyond the Wall to the north, there were the Picts, who constantly raided south into Roman territory. And beyond the sea to the west, there were the Irish (called then, "Scots"), who constantly raided the towns on the coast. The Picts were probably mostly looking for cattle to steal. The Irish came to kidnap people and sell them into slavery. One of the Irish captains captured Patrick, and sold him into slavery in Ireland. This was his first entry onto the Old Sod, and he was a most unwilling immigrant. Eventually, he escaped. The story of his call by God (via the voices in Focluit Wood) and his return can wait for another day.
My point is, Britannia was under constant threat, not only by the Picts and the Irish, but also by the Germanic raiders who hit the southeast coast -- the so-called Saxon Shore. The Roman military had its hands full beating back all these threats. But about the time of Patrick's birth, this got much harder, because the last regular Roman troops were withdrawn from Britannia to defend the central lands of the Empire (and to be wasted in civil wars between rival claimants to the imperial throne). Around the time that Patrick was a teenager, the Britons asked Rome to send troops back, because the raids had gotten worse. Rome replied, in AD 410, that they had no troops to send, and the Britons should see to their own defenses. Local militias did the best they could, but Britannia was under siege.
Eventually, one of their kings (Vortigern) had the bright idea of hiring the Saxons to fight off the Picts and Scots. This led to more Anglo-Saxons coming over and tearing off bits of the country for themselves. Eventually, it led to the creation of England (Englalond). The remaining British only lived under their own rule in the far west, where they were referred to as Wealas, the ancestor of our word, "Wales." It meant, "(romanized, celtic) foreigners."
Interestingly, this word, wealas, shows up in all the Germanic languages of the time as the word for Roman natives. It is the source of the name "Wallachia," for instance. And the Anglo-Saxons were not the only Germanic peoples flooding into the territory of the Empire to tear off their own little kingdoms. The Fifth Century is the time called by historians die Völkerwanderung, the age of migrations, of "people-wandering."
Beginning in the Third Century, the Germanic neighbors of Rome were increasingly important to the Empire. They served in its armies, when they weren't fighting its armies. And in the Fifth Century, all hell broke loose, as hordes of peoples on the march simply moved into the Empire, overwhelming all attempts to make them settle in designated places and live according to Roman law. Why did they come? In some cases, they were driven by even scarier people to their rear (like the Huns). But they were also drawn by the lure of opportunity. Rome offered a higher standard of living, more opportunity, more status and -- very important -- more security. They wanted these things; they just didn't want to have to become Roman.
So they flooded in and set up little kingdoms for themselves: Visigoths, Ostrogoths, Vandals, Suebi, Burgundians, Franks, Avars, and (later) Lombards. Some of them became fully independent right away, but others clung to a nominal allegiance to Rome. They were considered foederati, allied kingdoms within the Empire. In this capacity, they were sometimes a help: without the assistance of the Visigoths of Aquitania, Aetius could not have beaten Atilla at the Battle of Chalons (451). But soon enough, they simply tore the Empire apart, even while maintaining the fiction of Romanness.
All this happened during Patrick's lifetime. He outlived (probably) the last Roman Emperor in the West, who was deposed in 476. Patrick, of course, was fully occupied in Ireland, which had never been part of the Empire. His views on what was happening in the rest of Europe are not recorded. But it's important to realize that what happened during his lifetime was the destruction of the whole world he was born into. A new society -- for the time, a very dangerous society -- was being born.
We should not let our post hoc knowledge of how the Germanic invaders became the familiar, friendly nations of Europe today blind us to the staggering losses endured by Patrick's birth people and their fellow Romanized citizens. The burning, raping, pillaging, kidnaping, murder, and theft of lands was not a good thing for those subject to it. And we should learn the right lesson for today: that letting huge numbers of foreigners into your territory who have no intention of assimilating to your society (and who are, when allowed to be, capable of great violence) is not a smart idea.
Ask a contemporary of Patrick: the Fifth Century is the age of King Arthur (if he existed). The tragedy of his life is that, for all his successes, he could not save his people from being overrun after his death.