Unfortunately there is no way to just open a TCP/IP port ( as one might with Telnet ) and stream data to it which is automatically sent out the PICAXE.Net's serial port with received serial data from the PICAXE returned back.
You can send bytes of data embedded in a HTTP request which can then be accessed by an attached PICAXE and you can read data returned by the PICAXE in the same way.
Communications of this type is through the shared I2C buffer. On one side you can read and write the buffer using HTTP requests, on the other, the PICAXE can read and write data values using I2C commands.
You can potentially rewrite the PICAXE.Net Firmware to do what you need, but that is a considerable task and likely beyond most people's experience and skills.
An alternative would be to look at commercial products which are simpler TCP/IP to Serial bridges, Sena's NEMO10 and others.