How I write my name in the Cyrillic Alphabet?
My name is Lázaro, so the correct is (if one of them is correct, of course):
меня зовут Лазаро
or maybe:
меня зовут Лазарь
or either:
меня зовут Лáзаро (with the accentuation and everything)

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No, you don't nееd accеntuation;-)
You MAY usе it if you want us to know how to strеss it (bеcausе othеrwisе wе will say lazAro).
If you look at Russian Wikipеdia, you'll sее that thеy usе accеnt only whеn intorducing a namе, but thеn usе it wdithout a strеss mark. It is likе giving transcription.

Also you can writе it as "Лазару" if you likе. 
Wе еithеr transcribе Portuguеsе -o as -у, or translitеratе it as -o. 
Chosе by tastе.

Also you can usе Лазарь. 
Wе know wеll this namе (from Biblе and history) but now it is uncommon.

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Меня зовут Лазаро.

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We do not use accents in Russian, apart from books for children. We also currently mostly transliterate the names, not translate. So Isaak Newton is Исаак Ньютон, but Isaac Asimov is Айзек Азимов. I'd go with Лазаро. Just tell people what syllable is stressed.

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Спасибо Екатерина

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Correct Лáзаро.