Process.
The website should go through a design process before it even touches the developer and the UI/UX should be thoroughly thought out so that there are no complications when it reaches you and your team. Assuming that process has been followed then there isn't much for you to do other than develop it as being mindful for accessibility, user experience and how elements fit and relate to each other should have all been covered already.
In terms of actually developing the design then that's more or less down to you. Many agencies use different methodologies on how they approach this but I've always favoured developing a site in a preprocessing language like LESS/SASS and nesting my elements with inline media queries. That way you can see exactly how an element responds on the page without having to reference a seperate media query stylesheet, or scrolling traditionally to the bottom of the sheet to see a bunch of responses to a certain dimension.
For example below we can see that we have a square red box which at tablet resolution becomes slightly smaller and blue, eventually it becomes even smaller and pink at mobile:
#box{
height: 100px;
width: 100px;
background: red;
transition: 0.6s all;
@media screen and (max-width: 768px) {
background: blue;
height: 75px;
width: 75px;
}
@media screen and (max-width: 360px) {
background: pink;
height: 50px;
width: 50px;
}
&:hover{
background: lighten(red,10%);
transition: 0.3s all;
}
}
