I think most historians agree that there is plenty of evidence that Constantine converted to Christian at some point (although it's still unclear exactly when he converted, and for what reasons). But I have a different question: did the average person living in the empire actually know that he had converted?
I ask because the coins and inscriptions he left behind show no obvious references to Christianity. When, if ever, did the normal Roman citizens find out about his Christian beliefs? And if they never did find out, who was the first Emperor to be "openly Christian"?
Both of these are from his earlier days. The coin is from his time in Britain, still before his use of Christianity really took off. The inscription is on the Arch of Constantine, a much older arch that was "restored" by Constantine after the battle of the Milvian bridge. This is the early 310's AD. Constantine was still very much beginning with Christianity. The Milvian bridge was a major turning point in Constantine's life. As for the average Roman, it would not have taken long to realise. Prior to Constantine's "conversion" (I use the quotation marks because it was a much slower process than many believe; he didn't exactly jump in 100% after the Milvian Bridge), there was a massive, empire-wide persecution of the church and as soon as Constantine became Augustus in the west (i.e. after Milvian Bridge) he immediately ended the persecution. In 313, he issued the Edict of Milan, establishing toleration of Christianity. This, coupled with the fact that he refused to perform the sacrifice to Jupiter upon taking the city of Rome would absolutely have made people think twice. If this was not enough evidence, the council of Nicaea (where he sat enthroned among the attending bishops) in 325 would have likely ended any doubt that Constantine was now pretty serious about this new faith. Constantine did not formally convert until he lay on his deathbed, however. I don't think this would have made anyone think that he was less of a christian at that time, however. For more information on this topic, I would direct you to "Constantine and Eusebius" by Timothy Barnes. Also, I would seriously recommend looking at the Encyclopedia of Ancient Christianity edited by Angelo Di Bernardino. Eusebius' Life of Constantine will also provide background information. The source is skewed, but it can definitely show quite well what Christians thought of him. I would also suggest Lactantius, who is much more balanced. For the pagan side of things, check out Eutropius (just google him and you can find his work quite easily) and also Zozimus if you want an angrier pagan historian.