Origin of the phrase
There are actually a number of texts that are labeled the "Nicene Creed". The text produced by the 325 council does not include information about the kingdom at all. Eusebius of Caesarea, who attended the council, wrote back to his congregation about the deliberations. He reports an initial version of the creed that was used as a part of the baptism ceremony and the final version as issued by the council. None of these versions include the phrase "whose Kingdom shall have no end".
The Nicene Creed received by the Eastern Orthodox Church has traditionally been associated with the Council of Constantinople in 381. 36 representatives of Macedonianism were turned away from the second ecumenical council because they would not affirm the Nicene Creed (presumably the 325 version). The Council of Ephesus in 431 reaffirmed the faith established in Nicea, but seems to have had the 325 text in mind. Further, it seems to have put an end to the writing of new creeds. According to Wikipedia, the creed that included the phrase was not quoted until the fourth ecumenical council at Chalcedon in 451.
Philip Schaff commented on the origin of the phrase:
This clause had already, so far as the meaning is concerned, been added to the Nicene Creed, years before, in correction of the heresy of Marcellus of Ancyra, of whose heresy a statement will be found in the notes on Canon I. of this Council. One of the creeds of the Council of Antioch in Encæniis (a.d. 341) reads: “and he sitteth at the right hand of the Father, and he shall come again to judge both the quick and the dead, and he remaineth God and King to all eternity.”
This may be confirmed by Athanasius (On the Councils of Ariminum and Seleucia, II, 22) and Socrates Scholasticus (Historia Ecclesiastica, II, 10). For the purposes of this answer, I will assume the phrase dates back to at least the 341 Synod of Antioch.
Does the phrase reject Chiliasm?
The nature of the Second Person of the Trinity was the primary focus of these councils since heresies in question (those of Arius and Nestorius) were related to the meaning of the titles "Son of God" and "Son of Man". As far as I can tell, these heresies do not require a commitment to a particular eschatological theory.
If the phrase were added in response to Marcellus of Ancyra, who was accused of Sabellianism, the purpose would be to emphasize that Jesus would not return the Kingdom to the Father. In particular, "[Jesus] remains God and King to all eternity" would contradict dynamic monarchianism, which holds that God transferred his kingly authority to Jesus only for a time. Again, this view does not by necessity hold to a 1000-year reign; rather, it holds that God temporarily took on the persona of Jesus.
So the heresies the creeds were designed to contradict were not particularly millenarian. In addition, the phrase ought not particularly bother the believer in chiliasm. That's because unending Kingdom of Christ is said to come after the final judgement, which is not controversial. Revelation 20-22 describes a millenium in which the martyrs of the Church reign with Christ followed by the final judgment followed by the new Jerusalem where the servants of the Lamb with reign with Him forever. So the literal reading of Revelation meshes well with the phrase in the creed; the creed just leaves out some of the detail.
Did the fourth century church condemn Chiliasm?
The early church fathers are fairly well split on the issue:
Papias, Justin Martyr, Irenaeus, and Tertullian held views similar to the Jewish messianism of 1 Enoch, 2 Esdras, and 2 Baruch.
Origen, Dionysius of Alexandria, Eusebius, Jerome and Augustine preferred allegorical interpretations of apocalyptic literature.
However, the trend was from a more literal reading of the 1,000 years to a metaphorical reading. Perhaps there was no need to condemn chiliasm as a heresy since there were few proponents of the idea by the time the councils had solidified the Trinitarian position.
I did find an odd text while researching the question. During the 6th session, the Council of Chalcedon addressed the Emperor:
May God save your kingdom. [You] have put down the heretics, [you] have kept the faith. May hatred be far removed from your empire, and may your kingdom endure for ever!
It seems likely that the phrase "whose Kingdom shall have no end" addresses concerns over the nature of Christ and the incarnation, just as most of the other phrases that accumulated in the creeds. Chiliasm was losing favour by the 4th century, so it seems unlikely to have been targeted specifically by the creeds or councils.