Equation Array Exceed Right Margin

In addition to inserting some line breaks by hand in the final two rows, you also need to fix some of the notation by adding curly braces as needed in the terms q_11, q_21, q_22, etc. (And _k1 should be k_1, right?) When the parenthetic structures are nested, it helps readers if square brackets and curly braces are used in addition to round parentheses, to help detect the orders of the parentheses.

And you most definitely should not be using the badly deprecated eqnarray environment. Use the align environment, which is provided by the amsmath package, instead.

\documentclass{article}
\usepackage{amsmath} % for 'align' environment and '\notag' macro

\begin{document}
\begin{align}    
    k_1 &= hf(t_i, y_i)=hf_i \label{k1_RK_4}\\
    k_2 &= hf(t_i+p_1h, y_i+q_{11}k_1) \notag\\
     &= h(f_i+p_1hf_t+q_{11}k_1f_y) \notag\\
     &= h(f_i+p_1hf_t+q_{11}hf_if_y)\\   
    k_3 &= hf(t_i+p_2h, y_i+q_{21}k_1+q_{22}k_2)  \notag\\
     &= h(f_i+p_2hf_t+q_{21}k_1f_y+q_{22}k_2f_y) \notag\\
     &= h[f_i+p_2hf_t+q_{21}hf_if_y+q_{22}h(f_i+p_1hf_t+q_{11}hf_if_y)f_y] \notag\\    
     &= h[f_i+p_2hf_t+q_{21}hf_if_y+q_{22}hf_if_y+q_{22}h^2p_1f_tf_y+q_{11}q_{22}h^2\!f_i(f_y)^2]\\
    k_4 &= hf(t_i+p_3h, y_i+q_{31}k_1+q_{32}k_2+q_{33}k_3) \notag\\
     &= h(f_i+p_3hf_t+q_{31}k_1f_y+q_{32}k_2f_y+q_{33}k_3f_y) \notag\\
     &= h\{f_i+p_3hf_t+q_{31}hf_if_y+q_{32}h(f_i+p_1hf_t+q_{11}hf_if_y)f_y \notag\\
     &\qquad +q_{33}h[f_i+p_2hf_t+q_{21}hf_if_y+q_{22}hf_if_y+q_{22}h^2p_1f_tf_y \notag\\
     &\qquad +q_{21}q_{22}h^2\!f_i(f_y)^2]f_y\} \notag\\   
     &= h\{f_i+p_3hf_t+q_{31}hf_if_y+q_{32}hf_if_y+p_1q_{32}h^2\!f_tf_y+q_{11}q_{32}h^2\!f_i(f_y)^2 \notag\\
     &\qquad +q_{33}hf_if_y+p_2q_{33}h^2\!f_tf_y+q_{21}q_{33}h^2\!f_i(f_y)^2+q_{22}q_{33}h^2\!f_i(f_y)^2 \notag\\
     &\qquad+q_{22}q_{33}p_1h^3\!f_t(f_y)^2+q_{11}q_{22}q_{33}h^3\!f_i(f_y)^3\}
\end{align}
\end{document}

Besides the standard advice of not using eqnarray for whatsoever reason, see eqnarray vs align, I'd recommend using split for this, so to divide each block into a single unit.

With the tbtags option, we ensure that the number attached to a split equation is added at the bottom (when equation numbers are on the right) or at the top.

\documentclass{article}
\usepackage[tbtags]{amsmath} % for math

\begin{document}
\begin{align}    
k_1 &= hf(t_i, y_i)=hf_i \label{k1_RK_4}
\\
\begin{split}
k_2 &= hf(t_i+p_1h, y_i+q_{11}k_1) \\
    &= h(f_i+p_1hf_t+q_{11}k_1f_y) \\
    &= h(f_i+p_1hf_t+q_{11}hf_if_y)
\end{split}
\\
\begin{split}
k_3 &= hf(t_i+p_2h, y_i+q_{21}k_1+q_{22}k_2) \\
    &= h(f_i+p_2hf_t+q_{21}k_1f_y+q_{22}k_2f_y) \\
    &= h[f_i+p_2hf_t+q_{21}hf_if_y+q_{22}h(f_i+p_1hf_t+q_{11}hf_if_y)f_y] \\    
    &= h[f_i+p_2hf_t+q_{21}hf_if_y+q_{22}hf_if_y+q_{22}h^2p_1f_tf_y+
       q_{11}q_{22}h^2\!f_i(f_y)^2]
\end{split}
\\
\begin{split}
k_4 &= hf(t_i+p_3h, y_i+q_{31}k_1+q_{32}k_2+q_{33}k_3) \\
    &= h(f_i+p_3hf_t+q_{31}k_1f_y+q_{32}k_2f_y+q_{33}k_3f_y) \\
    &= h\{f_i+p_3hf_t+q_{31}hf_if_y+q_{32}h(f_i+p_1hf_t+q_{11}hf_if_y)f_y \\
    &\qquad +q_{33}h[f_i+p_2hf_t+q_{21}hf_if_y+q_{22}hf_if_y+q_{22}h^2p_1f_tf_y \\
    &\qquad +q_{21}q_{22}h^2\!f_i(f_y)^2]f_y\} \\
    &= h\{f_i+p_3hf_t+q_{31}hf_if_y+q_{32}hf_if_y+p_1q_{32}h^2\!f_tf_y+
       q_{11}q_{32}h^2\!f_i(f_y)^2 \\
    &\qquad +q_{33}hf_if_y+p_2q_{33}h^2\!f_tf_y+q_{21}q_{33}h^2\!f_i(f_y)^2+
     q_{22}q_{33}h^2\!f_i(f_y)^2 \\
    &\qquad+q_{22}q_{33}p_1h^3\!f_t(f_y)^2+q_{11}q_{22}q_{33}h^3\!f_i(f_y)^3\}
\end{split}
\end{align}

\end{document}