What would the altitude profile for a typical airline flight look like?
It is somewhat unclear what you want to achieve.

If you want to know how it looks like, you can search for some papers like this. To keep it short, altitude vs distance curve is pretty much similar to altitude vs time, but climb and descent segments are more curved with larger second derivatives for climb and descend. For details please read the paper.

If you are interested in some analysis, you need two components: flight data itself and data analysis software.

If you are looking for professional flight data analysis systems, for example, flightDataServices, generally they are not provided for free, but even if you have an access, you still need flight data for analysis. It can be more difficult to find it. If that is what you are looking for, just google "Flight Analysis Software" and "Flight Data".

You can do that analysis by yourself though, with a little less accuracy than a professional system. The closest option for getting an accurate flight data is FlightAware.com. They do not provide altitude vs distance out-of-the-box, but they provide plenty of data for further analysis. For example, this flight. You can open track log and see altitude/speed vs time (which is not you are looking for, but useful for a 'big picture'), and in the log itself you can find latitude, longitude, speed and rate of climb for some points. The points are provided with a less-than-a-minute frequency, which is pretty precise. It is not so hard to export them in any spreadsheet or CSV and generate whatever you are interested in: a pretty common data visualization task. It can be easily done with Matlab/Excel/LibreOffice. For example, altitude vs distance travelled, course vs time, rate of climb vs time, rate of climb vs distance etc.

If you are looking for data visualization software specified for flight analysis, I am not sure that something exists that fits all requests. Especially an online tool. The reasons are: flightaware provides both data and the most common plots (i. e. alt vs time and speed vs time); and if you already have data, it's pretty easy to plot whatever you want using Matlab/R/Python/Excel.