What are the fastest and easiest ways in KALIYUGA to get rid of LUST, ANGER, GREED,FALSE PRIDE and ENVY of ONE's SELF?
The easiest (and the only) path to Mokṣha in the age of Kaliyuga is the japa, bhajan, dhyāna, saṅkīrtana of Hari aka Lord Viṣhṇu.
(This answer is mainly from a Vaiṣhṇava literature-based perspective)
Lord Viṣhṇu says in Padma Purana: Uttara Khanda - 92.21/22 :
नाहं वसामि वैकुंठे योगिनां हृदये न च ॥२१॥
मद्भक्ता यत्र गायंति तत्र तिष्ठामि नारद
nāhaṃ vasāmi vaikuṃṭhe yogināṃ hṛdaye na ca ॥21॥
madbhaktā yatra gāyaṃti tatra tiṣṭhāmi nārada
“I do not live in Vaikuṇṭha, nor in the heart of the yogins.
O Nārada, I stay there where my devotees sing my praise.
So, Lord Viṣhṇu resides wherever his glory is sung by his devotees.
Further, the real significance of Hari-bhajan is expounded in the Bṛihana-Nāradīya- Purāṇa which states:
हरेर्नाम हरेर्नाम हर्र्नामैव केवलम्।
कलौ नास्त्येव नास्त्येव नास्त्येव गतिरन्यथा ॥ ३८.१२६॥
harernāma harernāma harrnāmaiva kevalam।
kalau nāstyeva nāstyeva nāstyeva gatiranyathā ॥38.126॥
The nama of Hari, the nama of Hari, the nama of Hari alone [is the only way].
In the age of Kali there is no other path to Liberation. ॥38.126॥
Vyasa says the following in Book 6: Chapter 2 of Vishnu Purana
कृते यजन् यज्ञैस्त्रेतायां द्वापरेऽर्चयन्।
यदाप्नोति तदाप्नोति कलौ सङ्कीर्त्य केशवम् ।। १७ ।।
kṛte yajan yajñaistretāyāṃ dvāpare'rcayan ।
yadāpnoti tadāpnoti kalau saṅkīrtya keśavam ॥17॥
"That reward which a man obtains in the Krita by abstract meditation, in the Treta by sacrifice, in the Dwápara by adoration, he receives in the Kali by merely reciting the name of Keśava." ॥17॥
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) at 12.3.51 also concurs with the greatness of Hari-naam japa to destory material bondages.
Hence, regular chanting of Hari-naam is a way to liberation.
यस्य स्मरणमात्रेण जन्मसंसारबन्धनात् ।
विमुच्यते नमस्तस्मै विष्णवे प्रभविष्णवे ॥६॥
Even a mere thought of Whom makes one liberated from the bondage of Samsara (birth and death) –
To that all-powerful Lord (Viṣhṇu), my profound salutations.
Further, in the same Sahasranāmam Yudhiṣṭhira questions Pitāmaha Bhīṣhma the same as follows:
किमेकं दैवतं लोके किंवाप्येकं परायणम् ।
स्तुवन्तः कं कमर्चन्तः प्राप्नुयुर्मानवाः शुभम् ।।२।।
(O Grandfather!) Who is the One Supreme Lord in this Universe? Who is One refuge for all? Who should be praised? And by worshipping Whom can a person reach auspiciousness (peace and prosperity)?
को धर्मः सर्वधर्माणां भवतः परमो मतः ।
किं जपन्मुच्यते जन्तुर्जन्मसंसारबन्धनात् ।।३।।
What is, in Your opinion, the greatest Dharma of all Dharmas? And by chanting Whose mantra, can a creature proceed beyond the bonds of Samsara?
And Pitāmaha Bhīṣhma replies as:
जगत्प्रभुं देवदेवमनन्तं पुरुषोत्तमम् ।
स्तुवन्नामसहस्रेण पुरुषः सततोत्थितः ।।४।।<br
He Who is the Lord of the Universe, Who is the God of Gods, Who is limitless,
and Who is the Supreme Purusha (Spirit); One who praises that Lord by the Sahasranama (the thousand names);
तमेव चार्चयन्-नित्यं भक्त्या पुरुषमव्ययम् ।
ध्यायन् स्तुवन्-नमस्यंश्च यजमानस्तमेव च ।।५।।
One who worships and prays, daily without break, that Purusha Who does not change (Who is imperishable) -
By meditating on Him, praising Him and making prostrations on Him;
अनादि-निधनं विष्णुं सर्वलोक-महेश्वरम् ।
लोकाध्यक्षं स्तुवन्-नित्यं सर्वदुःखातिगो भवेत् ।।६।।
To that Lord Who is beyond the six changes beginning with origin (birth) and ending with death, Who is the greatest Lord of all the worlds,
And Who is the knower of the world; one who worships that Lord by singing His name, such a devotee transcends over all the sorrows (duhka) of this material world (attains liberation) -
ब्रह्मण्यं सर्वधर्मज्ञं लोकानां कीर्ति-वर्धनम् ।
लोकनाथं महद्भूतं सर्वभूत-भवोद्-भवम् ।।७।।
By worshipping Him Who is the protector of the Brahma and the Vedas, Who is the knower of Dharma, Who increases the fame of all people,
Who is the Master of the Universe, Who is exceedingly wonderful, and from Whom the world along with all other beings have originated.
एष मे सर्वधर्माणां धर्मोऽधिकतमो मतः ।
यद्भक्त्या पुण्डरीकाक्षं स्तवैरर्चेन्नरः सदा ।।८।।
I consider this to be the most superior to all other forms of Dharma -
The Dharma which consists of unfailing adoration to that lotus-eyed Lord with the hymns of praise recited with devotion.
यतः सर्वाणि भूतानि भवन्त्यादियुगागमे ।
यस्मिंश्च प्रलयं यान्ति पुनरेव युगक्षये ।।११।।
He from Whom all beings have come into existence at the beginning of the cycle of time (creation),
And He into Whom all beings dissolve at the end of the cycle (dissolution);
तस्य लोकप्रधानस्य जगन्नाथस्य भूपते।
विष्णोर्नामसहस्रं मे शृणु पापभयापहम् ।।१२।।
He Who is the chief of the world, He Who is the Lord and King of the world,
And He Who is the remover of all sins and fear – hear from me, O King, the praise of that Lord which consists of the hymn of the thousand names, also known as “Vishnu Sahasranama”.
Thus, for one of the best methods to remove all kinds of ills/hardships from one's life and to end the endless samsara cycle (i.e., attain Liberation), both Veda Vyāsa & Pitāmaha Bhīṣhma concurs with the recital of the Viṣhṇu Sahasranāmam as the best path.
Furthermore, in the concluding part of the Sahasranāmam, Śhri Kṛiṣhṇa assures, that even a single name uttered with devotion is accepted by him:
यो मां नामसहस्रेण स्तोतुमिच्छति पाण्डव ।
सोहऽमेकेन श्लोकेन स्तुत एव न संशयः ॥२४॥
Lord Kṛiṣhṇa says:
He who likes to sing my praise, O Arjuna, using these thousand names, Know you that I would be satisfied by his singing of even a single stanza. There is no doubt on this.
So, to conclude, from the Vaishnavas perspective, The easiest & fastest (and the only) way to ameliorate all the material ills-hardships, and attain Liberation, in the age of Kaliyuga is the japa and bhajan of any of the names of Hari aka Lord Vishnu.
For Moksha in any yuga.:
When the unrighteous or sinful Kali Yuga comes, one should never pass a moment without devoting his heart upon Mahadeva. One that has drunk the Amrita constituted by the devotion to Hara, one becomes freed from the fear of the world (by getting moksha).(MBH 13.14).
For more you can read this answer of mine. Which is universal for all Shrutis ans Smritis period.
I hope this clarifies your queries.