If your coordinates are still in the csv file, you will want to read them in as a table
Table To Table—Help | ArcGIS Desktop
and create a feature class (can be in a file geodatabase and not necessarily a shapefile).
Adding x,y coordinate data as a layer—Help | ArcGIS Desktop (as mentioned by Wu)
There are many Lambert versions available, so you will have to see if the one you need is supported. When you are creating the XY layer, you will need to set the coordinate system for the input data, and you can search for "Lambert" to see you options.
Then I would follow Dan's advise on either creating a fishnet, or if these are centerpoints and not lower-left (or other corners), there might be other ways to show the 1km resolution.