I would recommend moving to regular tikz and pgfplots because they are way more flexible and also allow to tweak (for example) the background of tick labels. In both solutions presented here the some grid lines are penetrating the tick label space. If you are willing to switch, you could use the answer already recommended in the comment below your question.
If you want to stick to tkz-euiclide, you could use one of the two following solutions.
First Solution
This solution is cleaner than the second solution because only the part on the coordinate system that requires special treatment is changed. However this solution is more complicated and also has its issues.
\documentclass{article}
\usepackage{amsmath}
\usepackage{tkz-euclide}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\tkzInit[xmin=0,xmax=1,ymin=-1.5,ymax=1.5,xstep=.25,ystep=.5]
\tkzGrid
\tkzAxeY
\tkzDrawX
\tkzClip[space=0.1]
\begin{scope}[xscale=4/pi]
\tkzLabelX[trig=4,below=.25cm]
\end{scope}
\end{tikzpicture}
\end{document}
You can split \tkzAxeXY into its underlying parts \tkzAxeX and \tkzAxeY. The y-axis is not a problem, therefore we will concentrate on the x axis only. \tkzAxeX can be split into \tkzDrawX and \tkzLabelX, which is necessary because the axis line and ticks must be drawn separately from the axis ticks labels for this solution to work.
By splitting all this up we can draw a dummy coordinate system with xmax=1 and xstep=0.25, which will yield correct lines for the grid and the axis ticks, and utilize the trig=4 option only for the tick labels, which will yield correct labels at wrong positions. To correct the positions a scope with xscale=4/pi from normal tikz is used. (Notice that I used pi here instead of a number, this is a special constant that tikz and also tkz-euclide know.)
Unfortunately the internals of tkz-euclide cause a rounding error that make \tkzLabelX print one tick label to much. This excessive tick label is clipped away with \tkzClip.
Side note: If you want to draw points or other things into your coordinate system, remember that pi is located at 1.
Second Solution
I only stumbled upon this solution on my way to the first solution and I actually have no idea why exactly it works. I will still post it, maybe someone else knows how the parameters xmax and xstep in \tkzInit play together to allow this output.
\documentclass{article}
\usepackage{amsmath}
\usepackage{tkz-euclide}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}[xscale=4/pi]
\tkzInit[xmin=0,xmax=pi,ymin=-1.5,ymax=1.5,xstep=.999,ystep=.5]
\tkzGrid[xstep=0.25*pi]
\tkzAxeY
\tkzAxeX[trig=4,below=.25cm]
\end{tikzpicture}
\end{document}
In this case the whole picture has to be scaled in x direction, which also move the x axis label further to the right. This is why I personally consider this solution less clean than the first solution.
Side note: If you want to draw points or other things in this coordinate system pi is located at pi, which is nice.
