Cool, this is totally up my alley! What's also cool is that we have comparatively a lot of sources for a period of history and an area of the world that is characterised by a "frustrating lack of sources" (G. Rex-Smith).

It is very difficult to accurately determine the relationship between the personal inclinations of rulers and their actions even today, and this problem becomes much larger when we try to extrapolate it to events more than a millennium ago. Moreover, to what degree can we separate religion from politics, especially in pre-modern pre-Enlightenment societies? Even nowadays, there are serious discussions on the degree to which Islam as a belief and Islam as a political ideal can be separated. It's likely that Dhu Nuwās didn't particularly like Christians, but a lot more was going on. Let's go back to the unification of Yemen first:

So you have to imagine that the history of Yemen was characterised by four rival kingdoms: Saba, Hadramawt, Qahtaban and Mina. They were frequently at war, although it is obvious that for the majority of this period the kingdom of Saba was the most powerful. This changed around the 2nd century BCE, when they suffered a series of military setbacks and subsequently were seriously reduced. This, in turn, led to increased decentralisation of the lands still under their control which allowed for the rise of a fair amount of tribes called the Himyarites. Over the course of about four centuries(!), this confederation was able to amass so much power that they subjugated and annexed every other kingdom in the area, projecting their power even to what is now northern Saudi Arabia.

The rise of the Himyarites is significant, as it marked the first time in history that the entirety of Yemen was united under a single power. Moreover, the historical data indicates that Judaism had made faster and more successful inroads into South Arabian culture than Christianity. Across the straits of Aden, the Christian King of Axum had quite openly declared himself a "Servant of Christ" and laid claim on lands under Himyarite control. You understand that this led to some degree of animosity between the two.
So by the beginning of the 6th century CE, relations between Axum and Himyarite Yemen had turned rather sour. It's important to keep in mind that Dhu Nuwas' predecessor was not as particularly anti-Christian: although he executed some preachers for proselytisation, there was no indication of wide-spread persecutions. More importantly, he exhibited strong pro-Byzantine leanings, which was not acceptable for a large segment of the tribal aristocracy. Interestingly, Nebes notes that the largest portion of support for Dhu Nuwas came from Sabaean tribes whose influence had become diminished under Himyarite rule. Around the same time, Axum had already led a military intervention into South Arabia, which did not do much to alleviate tensions.
So by the time Dhu Nuwas actually came to power, relations between the Himyarites and Axum had deteriorated immensely. He proceeded to raze churches, massacre groups of Christians and basically tried to provoke the Axumites as much as possible. To his credit, this worked: the King of Axum led a full invasion, destroyed Dhu Nuwas' armies and reduced Yemen to a client state to be annexed some fifty years later after a final failed uprising.

 
TL;DR: Dhu Nuwas represented an anti-Axumite faction of Himyarite society and his actions were mostly political

Recommended readings

Hoyland, R. - Arabia and the Arabs
Nebes, N. - "The Martyrs of Najran and the End of Himyar" in The Qur'ān in Context, ed. Angelika Neuwirth.
Robin, C. - "Quelques épisodes marquants de l'histoire de l'arabie du sud"
–––––– "Himyar et Israël"
–––––– "Du paganisme au monothéisme au Yemen"