Is determining the volume of the contaminated soil the ultimate goal or do you need to visualize this? You need to build something like an isosurface or a 3D voxel grid from your input data then determine its volume which is something 3D analyst cannot do.
If you just need a rough model then you can create sort of a layer cake multipatch model in 3D by extruding between surfaces to obtain a multipatch geometry for each level. In geology software for example a lot of interpretation goes into creating subsurface models so using a similar approach you could separate your data into groups of points by elevation then using the sample points as a visual guide draw approximate polygons defining the outer most extent of the contamination for each elevation group (every x metres). Each time you complete a polygon for one level you asign it a height field. What you end up with will be a series of flat polygons stacked on top of each other when you view the results in ArcScene. You can load in your points at the same time and QAQC your results. The stacked polygons should already resemble the shape of the plume. Next step is to use the extrude between tool to create a 3D multipatch for each level. when you do this for all the levels and add it to ArcScene you should basically have a rough 3D model of the plume. You can use the Union 3D tool to merge the layer cake together into a single model. Your multipatch should be closed so you can use the add Z information tool to obtain volume. This is essentially a similar process to how subsurface solid object are created in geological software except there you can digitize in real 3D space (snapping to drillhole assays, points in space or whatever else is loaded) and create your polygon slices inclined and in whichever direction. You can then connect the slices with directional polylines in 3D space to "tell" the software in which direction to build the mesh.
In truth, 3D analyst is not the tool for subsurface modelling. There is no actual way to digitize multipatches from scratch in 3D space. You can import multipatches from 3D DXF created in datamine studio, Gemcom, Leapfrog, Geosoft, etc. but then what? You can't label anyting in 3D, you can't produce a decent vertical section, you can't import drillhole data or borehole logs which is how most subsurface data is obtained. I didn't even find an easy way to set a specific plunge and azimuth in arcscene.
There are other ways you could build the multipatch such as using Google Sketchup - Collada. The problem is that in version 10 you can only export collada to Sketchup from ArcGIS. First you have to create a multipatch then convert to Collada then export to Sketchup. This is only an OK process for buildings and such - you export boxes and make them look like buildings in Sketchup then bring them back to ArcScene. Up to version 9.3.1 you were able to use the Google Sketchup plugin for ArcGIS which allowed you to export points, polygons, polylines and TINs as well so you could have technically exported your point cloud then make a 3D object of the plume in sketchup and then bring it back. Georeferencing was an issue though so you needed to export control points with then georeference the model back to these control points. Still i used to do it a lot. Gone are the days were i was able to easily export a complex surface model from TIN to sketchup without any issues. The Collada way is too buggy. It only handles simple polygons and you are lucky if something does not die during the process. So it really only works for buildings.
You might find a lot better answers on GIS StackExchange forum. Most of the GIS wizzards that used to frequent the ESRI forums have moved there and there are some serious experts there. In my experience questions that have to do with interpolation or 3D modelling receive a lot of attention.
