"Clovis received an appointment to the consulship from the emperor Anastasius [491-518], and in the church of the blessed Martin he clad himself in the purple tunic and chlamys, and placed a diadem on his head. Then he mounted his horse, and in the most generous manner he gave gold and silver as he passed along the way which is between the gate of the entrance [of the church of St. Martin] and the church of the city, scattering it among the people who were there with his own hand, and from that day he was called consul or Augustus. Leaving Tours he went to Paris and there he established the seat of his kingdom." - Gregory of Tours, Histories 2.38
Gregory of Tours portrays Clovis as very 'Roman'; he's a Catholic, he wears a lot of purple in this passage and later Clovis commissioned the first Salic Laws. So what was his relationship to his Roman subjects like?
edit: Seems I've semi-resolved this myself
It seems Clovis was on fairly poor terms with catholics in his kingdom until his victory against Alaric II in 507.
Many catholics believed 'barbarians' should be segregated from more civilised catholicism (largely as Arianism was the dominant religion of the Visigoths in Toulouse, Vandals in Carthage and Ostrogoths under Theodoric). Alaric II re-issued the Theodosian Code, banning Arian conversion to Catholicism - many of the Cathlolic priests in Clovis' kingdom lauded him for this;
"with our knees bent to the ground, we prayed for the kingdom of Alaric ... so that the Lord might expand his realm"
However, Clovis was determined to become catholic; victory over the Alemanni had already convinced him of the power of Christ. The bishop of Reims then baptised Clovis and his entire 3,000 man army, many of whom were renegade Roman soldiers. Clovis then sought an oracle from St. Martin at Tours in order to gain divine favour over a war against Alaric;
"'Lord God,' said he, 'if you are on my side ... deign to show me a porpitious sign as these men enter the church, so that I may know that you will support your servant Clovis'"
So winning the war in 507 resulted in the very catholic precession outlined by Gregory of Tours, in another attempt to gain the favour of Roman catholics.
I would argue that Clovis had pretty good relations with his Roman subjects. From his ascension, Clovis had to deal with Romans - sometime in the 480s, Clovis received a letter from Remigius, Bishop of Rheims, advising the young ruler on how to govern Belgica in a very Roman fashion. In Ian Wood's words:
Whatever date one ascribes to the letter, Remigius conceived of Childeric's [Clovis' father] power in terms of Roman provincial rule, and he also thought that the clergy had a right to advise, even though the ruler might be barbarian and pagan.
Others take it further, such as Edward James, who essentially argued that Clovis had inherited the governorship over Belgica from his father. This is probably going to far, but I think it would be a mistake to suggest that the 'barbarians' were isolated from the Romans. In fact I would argue the opposite and instead say that the 'barbarians' loved everything Roman and did their best to maintain Roman institutions. Remigius is an excellent example, as the bishop and the young king seemingly formed a lasting friendship, as we also have a letter from the bishop offering condolences to the king after his sister's death. Moreover, it was Remigius who finally baptised the king after his conversion. The line you quoted, about the bishops praising Alaric, was from a council held in Agde, deep within the Visigothic kingdom, so it was in fact local bishops loudly proclaiming how awesome the 'Arian' king was for allowing Nicene Christians to gather for a church synod. As with the division between 'barbarians' and Romans, we shouldn't emphasise the division between 'Arians' and Nicenes too far either.
Secondly, it is worth noting that Gregory of Tour's much later account of Clovis' conversion is deeply flawed. From a contemporary source, a letter by Bishop Avitus of Vienne, it is implied that Clovis had already flirted with 'Arianism' before finally settling on converting to Nicene Christianity instead - this is not that implausible, as Clovis' sister was an 'Arian' at this point even according to Gregory of Tours. The political situation around Clovis' conversion, which occurred sometime between 506 and 508, was also much more complicated, as Clovis' attack on the Visigoths can be linked to Emperor Anastasius' policy of weakening Ostrogothic Italy, the Visigoths' ally. Indeed, even as the Franks fought the Visigoths, the emperor's ships began harassing the Italian coast. From this perspective, Clovis was perhaps awarded the title of consul either as a reward after the campaign or given the title beforehand (the chronology is still debated). Likewise for the title of augustus, a title that has recently been suggested to be derived from a genuine Roman title, the augustal prefect who governed Egypt; perhaps Gregory of Tours, writing generations later, misreported a similar title that was granted to Clovis. Or, just as likely, Clovis had grand pretensions and saw himself as a new augustus, which wasn't so odd at the time, as Theoderic the Great of Italy had also been proclaimed by his Roman subjects as an augustus, even though he was an 'Arian' 'barbarian'.
Clovis was in fact behaving no differently from his contemporaries. Like Theoderic in Italy, Clovis built a powerful kingdom backed by Roman institutions. Like Gundobad in Burgundy, he received grand titles from the emperor. Like Alaric II, he issued laws in the Roman mold. The post-Roman world was certainly different from the empire at its height, but it was still Roman in many respects and the emperor, though he lived in distant Constantinople, still had a great deal of influence in the west.
For a reassessment of Clovis' conversion, D. Shanzer's 'Dating the baptism of Clovis: the bishop of Vienne vs the bishop of Tours', Early Medieval Europe, 7 (1998), is essential. More recently, R. Mathisen et al (eds.), The Battle of Vouillé, 507 CE: Where France Began (2012), contains an excellent collection of essays on a subject you seem to be interested in, noting in particular the roles played by Anastasius I and Theoderic the Great in Gallic affairs. General overviews, such as I. Wood, The Merovingian Kingdoms (1994), all also mention how Roman institutions were still harnessed by the 'barbarians'. Gregory of Tours is a wonderful source, but he has to be read carefully and I think it would be fair to say that modern historians, particularly since the publication of W. Goffart, The Narrators of Barbarian History (1988), now view him very differently.