Why did Lord Chaitanya appear on earth? Is he a Vishnu Avatar?
Harinama Sankirtana (chanting of the Lord's names) is the prescribed spiritual duty for the people of Kali Yuga:
kṛte yad dhyāyato viṣṇuṁ tretāyāṁ yajato makhaiḥ
dvāpare paricaryāyāṁ kalau tad hari-kīrtanāt [SB - 12.3.52]
Whatever result was obtained in Satya-yuga by meditating on Viṣṇu, in Tretā-yuga by performing sacrifices, and in Dvāpara-yuga by serving the Lord’s lotus feet can be obtained in Kali-yuga simply by chanting the name of Hari.
Although the scriptures defined the prescribed duty, they didn't tech how to do it practically. For this pupose a teacher is required. And who could other than the God himself can teach us how to practice His nama samkirtana? So God from time to time takes birth here for teaching us the practical process:
martyāvatāras tv iha martya-śikṣaṇaṁ
rakṣo-vadhāyaiva na kevalaṁ vibhoḥ [SB 5.19.5]
But God takes avatars here to teach or educate the people, not only just to kill the demons.
It is for this reason that Lord Chaitanya took birth here. He revived the samkirtana movement, brought into public the Hare Krishna mahamantra, and staying as a sannyasi Himself, He taught others how to practice devotion through the process of chanting the Lord's names and forms. In those mediaval times irreligious activities had taken over and whether or not Lord Chaitanya's divinity is accepted, His activities and pastimes restored the yuga dharma and many people are getting the opportunity of chanting the Lord's names now.
So the Chaitanya Charitamrita states Shir Krishna's desire for taking birth as a devotee:
yuga-dharma pravartāimu nāma-saṅkīrtana
cāri bhāva-bhakti diyā nācāmu bhuvana [CC - 1.3.19]
I shall personally inaugurate the religion of the age — nāma-saṅkīrtana, the congregational chanting of the holy name. I shall make the world dance in ecstasy, realizing the four mellows (dāsya, sākhya, vāstalya, mādhurya) of loving devotional service.
Regarding Him being an avatara of Vishnu, people have different views. But in Gaudiya Vaishnavism, He is accepted as an incarnation of Vishnu or Shri Krishna [CC - 1.5.6]. Regardless of His divinity, He is revered as a spiritual teacher by whose mercy one crosses this material ocean filled with crocodiles of various opinions:
śrī-caitanya-prabhuṁ vande bālo ’pi yad-anugrahāt
taren nānā-mata-grāha-vyāptaṁ siddhānta-sāgaram [CC - 1.2.1]
I offer my obeisances to Sri Caitanya Mahāprabhu, by whose mercy even an ignorant child can cross the ocean of theories filled with crocodiles of various opinions.
Chaitanya Mahaprabhu is only thought of as an incarnation of Vishnu by the Gaudiya Vaishnava sect, which includes the organization ISKCON. Other Vaishnava sects, like the Sri Vaishnava sect I belong to, don't regard him as an incarnation of Vishnu. In any case, here is a verse in the Srimad Bhagavatam that Gaudiya Vaishnavas consider a prophecy of Chaitanya:
In the Age of Kali, intelligent persons perform congregational chanting to worship the incarnation of Godhead who constantly sings the names of Kṛṣṇa. Although His complexion is not blackish, He is Kṛṣṇa Himself. He is accompanied by His associates, servants, weapons and confidential companions.
Note that this is a translation by a Gaudiya Vaishnava, so it's written from the point of view that it is a reference to Chaitanya. The translation by J.M. Sanyal (not online) words things somewhat more neutrally:
The wise, with chanting songs, celebrating the name of The Lord, adore him who is black, and with his body and limbs, ornaments, weapons and attendants like Garuda.
Gaudiya Vaishnavas also believe that this line from the Vishnu Sahasranama refers to Chaitanya:
He that is of golden complexion; He whose limbs are like gold (in hue); He that is possessed of beautiful limbs; He whose person is decked with Angadas made with sandal-paste; He that is the slayer of heroes.
According to the Mahajanas of the Gaudiya Vaishnava sect,
“The first purpose [of Krishna’s descent as Shri Chaitanya] was to relish the position of Śrīmatī Rādhārāṇī, who is the prime reciprocator of transcendental love of Śrī Kṛṣṇa. Lord Kṛṣṇa is the reservoir of transcendental loving transactions with Śrīmatī Rādhārāṇī. The subject of those loving transactions is the Lord Himself, and Rādhārāṇī is the object. Thus the subject, the Lord, wanted to relish the loving mellow in the position of the object, Rādhārāṇī.
“The second reason for His appearance was to understand the transcendental mellow [experience or taste] of Himself. Lord Kṛṣṇa is all sweetness. Rādhārāṇī’s attraction for Kṛṣṇa is sublime, and to experience that attraction and understand the transcendental sweetness of Himself, He accepted the mentality of Rādhārāṇī.
“The third reason that Lord Chaitanya appeared was to enjoy the bliss tasted by Rādhārāṇī. The Lord thought that undoubtedly Rādhārāṇī enjoyed His company and He enjoyed the company of Rādhārāṇī, but the exchange of transcendental mellow between the spiritual couple was more pleasing to Śrīmatī Rādhārāṇī than to Śrī Kṛṣṇa. Rādhārāṇī felt more transcendental pleasure in the company of Kṛṣṇa than He could understand without taking Her position, but for Śrī Kṛṣṇa to enjoy in the position of Śrīmatī Rādhārāṇī was impossible because that position was completely foreign to Him. Kṛṣṇa is the transcendental male, and Rādhārāṇī is the transcendental female. Therefore, to know the transcendental pleasure of loving Kṛṣṇa, Lord Kṛṣṇa Himself appeared as Lord Chaitanya, accepting the emotions and bodily luster of Śrīmatī Rādhārāṇī.”
Next we explore the more widely known reason for Shri Chaitanya’s appearance. Actually this “exoteric” reason seems quite esoteric to most of us, but it is in relationship to the suffering jiva souls like ourselves and it is considered external to the Lord’s personal inner purpose. Generally Krishna entrusts the deliverance of the fallen souls to his Vishnu part, since Krishna is mainly absorbed in his loving relationships with his pure devotees. Never the less, the Lord can accomplish many purposes simultaneously.
It is said in the Chaitanya Charitamrita that as Shri Krishna was contemplating his inner reasons for again coming to the world, it coincided with the time of the Yuga Avatar, or that incarnation who appears in every age to give the process of God realization. Most of us will appreciate this aspect more in the beginning since by the Lord’s grace we have a process of self-realization that is both easy and sublime, and yet gives us a benefit rarely available—namely pure love for Krishna in the mood of the inhabitants of Krishna’s confidential abode, or Goloka Vrindavan.
The apparent, or external, reason for the advent of Lord Chaitanya is given as follows: “Lord Śrī Kṛṣṇa, after displaying His pastimes as Lord Kṛṣṇa, [5,000 years ago] thought it wise to make His advent [again] in the form of a devotee to explain personally the transcendental mellows of reciprocal service and love exchanged between Himself and His servants, friends, parents and fiancées. According to the Vedic literature, the foremost occupational duty for humanity in this Age of Kali is nāma-saṅkīrtana, or congregational chanting of the holy name of the Lord. The incarnation for this age especially preaches this process, but only Kṛṣṇa Himself can explain the confidential loving service performed in the four principal varieties of loving affairs between the Supreme Lord and His devotees. Lord Kṛṣṇa therefore personally appeared, with His plenary portions, as Lord Chaitanya.”
It should be noted that Chaitanya “Mahaprabhu” (greatest master) is not an incarnation of Krishna or Vishnu as are the normal avatars, but since He is Krishna himself, he is also the original Supreme Personality of Godhead. Krishna’s world, Goloka Vrindavana, is compared to a lotus flower, and within that abode, is another lotus (the lotus within the lotus), which is the place of Shri Chaitanya and his associates, or Navadhvipa.
Chaitanya mahaprabhu rejected being called as supreme at many places..
One example is:
prabhu kahe, — ‘viṣṇu’ ‘viṣṇu’, āmi kṣudra jīva hīna jīve ‘viṣṇu’ māni — ei aparādha-cihna jīve ‘viṣṇu’ buddhi dūre — yei brahma-rudra-sama nārāyaṇe māne tāre ‘pāṣaṇḍīte’ gaṇana (chaitanya charitamrita madhya 25.78-79)
Meaning: Chaitanya mahaprabhu said , "I am insignificant living being bereft of any good qualities. Considering jiva to be vishnu is a great aparadha. To say nothing of ordinary living entities, even Lord Brahmā and Lord Śiva cannot be considered on the level of Viṣṇu or Nārāyaṇa. If one considers them as such, he is counted among pasandis"
Thus, if you don't consider him an avatar, then why he appeared doesn't arise.