making a preexisting site responsive

There's no magic solution for this.

Make up a grid then swap out each element for a grid class and remove it's positioning properties (float, margin, padding).

I wouldn't worry about a grid. I don't use grids when building sites and my sites are responsive and adaptive.

You just need to go through the site, working your way down in resolution and everytime the site looks rubbish, examine and then add something, be it change in sizes in CSS or a media query, to make it look good again.

Start by converting all the container sizes to percentages, working on how you'll rearrange them as the viewport gets narrower.

It depends on what you want to achieve. Without a grid, it's hard to really put up a clean site and make it responsive.

I would say it's nearly impossible, to put a site that doesn't have a grid, into a grid without making slight changes to its appearance. Now it depends on how complex the site is, to make it responsive without putting it in a grid. I would stick to what Matthew told you. If you see the browser, or rather the resolution of your monitor as the parent element, you can just use a "responsive calculator" to change the "px"-values to "%"-values. And everytime you reach a point where your site doesn't look good anymore you just add a media query for that resolution. A grid makes that process easier but it's possible without it too.