In the early Vth century, Christianity was not just a spiritual or social choice but also political, since Constantine and even more so after the death of Julian that saw the triumph of Imperial Christianity.This triumph was illustrated, and made possible, by the inclusion of the new religion in the social codes of a redefined elite that identified itself to public, thus imperial, service. This Roman state wasn't exclusively Christian (and tolerated, willy-nilly, the other religions) but the imperial and the religious values were interconnected, therefore serving the Empire was serving Christ, and that influed over the perceptions and relations the Empire had with Barbarians : Constantine having defeated Gothic peoples along the Danube stressed that bishops
"couldn't deny that he was the true servant of God, as my piety did so that everything is peaceful; Barbarians themselves that thus far were ignorant of the Truth, now know God trough me, His servant, praise His name as it is appropriate and fear Him, as events made them acknowledge that God was everywhere my shield and my providence; they fear us because they fear God"
Barbarians, in the IVth century as they did before, were in relation with the emperors in order to gain some acknowledgement, subsides, preferential treatment, military service, etc. either as junior partners or as defeated parties : as these relations had an important personal aspect, the religious matter was quite important. Pretty much as nobody was forced to convert to Christianism in Romania, the emperors did not forced Barbarians to convert (especially as Roman bishops and monks rather felt the need to convert the urban and rural populations of the Empire) : it was nevertheless a convenient way to integrate oneself to the empire, as a people, a band or even ambitious individuals.There were probably other reasons and motivations, such as the capture of Christian slaves or the desertion of Romans "passing to Barbarians" and wasn't necessarily contradictory with genuine religious experience, not unlike what happened during most of the Early and High Middle-Ages with the conversion of peoples outside the former Romania (Anglo-Saxons, Norse, Slavic peoples, etc.) or even, closer to us, the ideological polarizations of the XXth century you probably had a mix of genuine choice and opportunism.
(taking from this earlier answer)
The first to ostensibly adopt Christianity were part of the Tervingi Goths, a confederation of peoples that were particularily active in the mid-to-late IVth century at the northern-eastern border of the Empire. We don't know the details of what happened, and altough tradition ascribes their conversion to the Homean Creed^(1) to the heresy of their evangelizer Ulfila (an Anatolian slave taken by raiders), we don't know if Ulfila was actually Nicean, Homean or anything else : rather, these Goths seem to have converted to the creed that was then in favour at the court of Valens, the eastern Roman emperor, communities and their leader paying lip-service to Constantinople.
This was not yet, however, everyone's choice : Franks, already integrated within western roman frameworks , apparently felt little need to convert (at least, as political and military groups, especially as the western army might have been more religiously conservative than in the East). Even among the various Gothic groups and the Tervingi in particular, conversion represented not only a political choice towards the Empire but also in the constitution of opposed factions of communities : Athanaric and Fritigern, both petty-kings of the Tervingi, both acknowledging Roman suzerainity, fought each other for dominance the latter being Christian, the former being fiercly Pagan.
As Goths moved south of the Danube in the mid-to-late 370's, a series of vexations and conflicts let to the Battle of Adrianople : regardless of the other military and political outcome, it had two main consequences for the matter at hand. The defeat and the likelt death of Valens put an end to the imperial favour to the Homean Creed and its replacement by the Nicean Creed under Theodosius; but as well a formidable prestige gain by Goths that hold on to what had become an heresy not longer as subservience to the emperor, but to stress their distinctiveness once they settled as a largely autonomous group (half polity, half army) within the Roman Empire. As the theological and liturgial differences with Nicean Christianity were not only meager, but as Romans still were focused on dealing with paganism and heresy among Romans first, they were left with this convenient way to cultivate their difference.
In the same time, Barbarian communities settled within the Empire (especially since the IIIrd century) were themselves more or less importantly set in Roman frameworks : either led by Roman officers or their own petty-kings, with seemingly less agency (for the time being), and altough details are lacking at the very best, it is not unreasonable to argue they might have been Christianized and thus allowed to potentially move up the ladder of Roman military hierarchy all the easier.
The late IVth and early Vth century saw a double movement of conversion of Barbarian peoples : Burgundians, fleeing the Huns and entering Romania, thus converted to Nicean Christianity hoping that "the God of Romans" (and thus, the Empire) would provide with land and protection; whereas Goths went trough a prestigious missionary policy towards other Barbarian peoples entered in Romania, establishing thus relations of patronage and quasi-familial hierarchy trough symbolical god-fathering : we don't know exactly when Vandals converted to Christianism, but they seem to have done so by the 400's already and directly onto its Homean variant.
Indeed, the breakout of the limes in the mid-400's and the deep militaro-crisis the empire went trough did not removed a felt need to Christianize, quite the contrary : in a context of concurrential warlordism among Barbarians, adoption of Christianity (either Nicean as Suebi, or Homoean as the vast majority) still represented a display of romanitas from the petty-king and of integration within Roman social codes as they were fighting for honors, titles, grants and subsides.The two set of peoples that didn't convert, the Franks and Alamans, were incidentally fairly peripheral to the Vth century events and Late Roman/Barbarian politics.Even Homean Creed, whose heresy was tolerated as "they don't know better", was a very convenient way to distinguish 'Romans' from 'Barbarian' as groups as Goths or Vandals (themselves largely romanized) growingly included more and more Romans : fiscal or military deserters, slaves, peasants, all the rabble a prestigious and semi-moving army can attract, etc. As much as more or less superficial knowledge of Gothic, alliegence to the leader as king and general, adoption of militarized social codes, Homean Christianity became a marker of many Barbarian identities rather than what remained of traditional religion in this situation.
Political opportunism, cultural conveniance, identitarian necessities, but as well Roman religious laisser-faire and carelessness when it came to religious policies other than on Romans etc. were instrumental into leading to the conversion of Barbarians as they entered the Late Roman Empire. The fall of the western Roman state, however, had the consequence having heretic Barbarians ruling over orthodox Romans by the mid-to-late Vth century, an uneasy situation that lead by the latter decades to a more virulent denunciation of "Arianism" but also in some successful and opportunist networking from peoples that thus far didn't saw fit to convert : you might find some elements of answer onto the conversion of Franks in the early VIth century in the second part of the aformentioned answer.
^(1) While the Homean Creed is often, and contemporarily by its detractors, called 'Arianism' it's not as much anti-trinitarian than subordationist. Altough theological differences do exist, they're relatively and purposefully minor and didn't (at least until the VIth) led to a fierce theological opposition.