The article document class doesn't recognize -- and thus doesn't do anything with -- the option 60pt. The only three font size options the article class recognizes are 10pt, 11pt, and 12pt. Other document classes -- such as memoir and the KOMA-Script classes -- recognize many more font size options; however, to the best of my knowledge they don't recognize the option 60pt either.
To change the font size within a LaTeX document, use the command sequence
\fontsize{<nominal font size>}{<baselineskip>}\selectfont
Applied to your example, assuming you want a baselineskip that's 20 percent than the nominal font size, one could generate the following sequence of progressively larger font size (using a geometric progression with a factor of 1.2):

\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[bulgarian]{babel}
\usepackage{lmodern} % for better support of Cyrillic glyphs
\begin{document}
Ляляляля!
\fontsize{12}{14.4}\selectfont
Ляляляля!
\fontsize{14.4}{17.3}\selectfont
Ляляляля!
\fontsize{17.3}{20.7}\selectfont
Ляляляля!
\fontsize{20.7}{24.9}\selectfont
Ляляляля!
\fontsize{24.9}{29.9}\selectfont
Ляляляля!
\end{document}