The artist urgently needed an opportunity to present .....

...a way of presenting his works

In your example, either 'opportunity' or (less formally) 'chance' are possible;
'way' is not possible--although grammatically correct in that position, it alters the meaning.

Definitions from standard dictionaries explain the nuances, and thesauruses do a good job as well. Note especially the description under SYNONYMS of 'opportunity' from the American Heritage Dictionary:

Selected definitions from m-w.com:
possibility
1 : the condition or fact of being possible
2 ...
3 : something that is possible

opportunity
1 : a favorable juncture of circumstances <the halt provided an opportunity for rest and refreshment>
2 : a good chance for advancement or progress

chance
1 ...
2 : a situation favoring some purpose : OPPORTUNITY <needed a chance to relax>

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Excerpts from the American Heritage dictionary:

possibility: http://www.bartleby.com/61/56/P0465600.html NOUN 1. The fact or state of being possible. 2. Something that is possible. 3. possibilities Potentiality for favorable or interesting results: The idea has great possibilities.

chance: http://www.bartleby.com/61/57/C0235700.html 4. A favorable set of circumstances; an opportunity: a chance to escape.

opportunity: http://www.bartleby.com/61/82/O0098200.html NOUN: Inflected forms: pl. op·por·tu·ni·ties
1a. A favorable or advantageous circumstance or combination of circumstances. b. A favorable or suitable occasion or time. 2. A chance for progress or advancement.
SYNONYMS: opportunity, occasion, opening, chance, break These nouns refer to a favorable or advantageous circumstance or combination of circumstances. Opportunity is an auspicious state of affairs or a suitable time: “If you prepare yourself . . . you will be able to grasp opportunity for broader experience when it appears” (Eleanor Roosevelt). Occasion suggests the proper time for action: an auspicious occasion; an occasion for celebration. An opening is an opportunity affording a good possibility of success: waited patiently for her opening, then exposed the report's inconsistency. Chance often implies an opportunity that arises through luck or accident: a chance for us to chat; no chance of losing. A break is an often sudden piece of luck, especially good luck: got his first big break in Hollywood.

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From Roget's thesaurus:

possibility: (Roget's): http://www.bartleby.com/62/47/P1154700.html NOUN: 1. The likeliness of a given event occurring: chance, likelihood, odds, probability, prospect (used in plural). See LIKELY. 2. Something that may occur or be done: contingency, eventuality.

chance: http://www.bartleby.com/62/71/C0237100.html 4. A favorable or advantageous combination of circumstances: break, occasion, opening, opportunity. Informal : shot. See LUCK.

opportunity: http://www.bartleby.com/62/49/O1064900.html NOUN: A favorable or advantageous combination of circumstances: break, chance, occasion, opening. Informal : shot. See LUCK.

Online dictionaries such as American Heritage and Roget's Thesaurus (both at bartleby.com) and Merriam-Webster's Dictionary (at m-w) are useful references for working out questions like this one.