How to make italic greek letter \lambda

Latin Modern math doesn't blend with Times New Roman. You're better using NewTX:

\documentclass{article}
\usepackage{newtxtext,newtxmath}

\begin{document}

The text is in Times; $\lambda\lambdaup$.

\end{document}

If you insist in using Latin Modern math symbols, here's how you can do:

\documentclass{article}
\usepackage{amsmath}
\usepackage{lmodern}     % set math font to Latin modern math
\usepackage[T1]{fontenc}
\renewcommand\rmdefault{ptm} %change text font to Times New Roman

\DeclareSymbolFont{myletters}{OML}{ztmcm}{m}{it}
\DeclareMathSymbol{\uplambda}{\mathord}{myletters}{"15}

\begin{document}

\begin{align}
\textrm{Lambda in Latin modern:} \lambda \\
\textrm{Lambda in Times New Roman:}\uplambda
\end{align}
\end{document}

If you are able to use Lua- or XeLaTeX, I would recommend the package unicode-math:

% arara: lualatex

\documentclass{article} 
\usepackage{unicode-math}
\usepackage{fontspec}
\setmainfont{Times New Roman}
\usepackage{blindtext}

\begin{document} 
\blindtext
$\mathup{\lambda}\lambda$
\end{document}