What you want to do is a trivial programming task once you have decided a few things:
These are probably the three biggest questions you need to answer. I added some commentary, but look at each of these questions beyond this post to find what works for you.
Who do you want to use for your map? Since you only have one type of data you will want to display that data on someone else's nice looking map. The big choices are Bing, Google, OpenLayers/OSM, and ESRI. Your choice will most likely be driven by the licensing of the above services and if you are willing to pay or not. A need to support mobile devices may also factor into your decision. Since the map is what your users will see, choose the best looking map you can afford.
How will you serve up your data? You have several options to serve your speed limit data. GeoServer and MapServer and ESRI are some popular mapping software packages. If you only displaying a few layers of data all mapping software will be overkill. The actual software to render your map data will most likely affect only your pocket book, so free is good here usually.
Tiles vs Lines You will server your data as either a group of lines sent to the browser, or as pre-rendered tiles to be loaded on top of the map. If you data changes frequently you will want to serve it dynamically as line data (an array of points.) If your data does not change frequently, you should consider tiling your data. Tiling involves pre-rending of the entire map at all zoom levels. This allows the map to be loaded very fast and this how almost all base maps are rendered. The downside is that the tile generation can take a long amount of time and tiles can take a large amount of space.