You are almost there - cool.
What you have to do is when you check with serial.available that there is data then you should use a loop to read all the data one byte at a time until serial.available returns false. Put these data together to make a string, send that string to the LCD, and then use a small delay in your main loop to allow all the data the PC is sending in "the next batch" to be recieved before checking with seriel.available again.
The problem is that serial.available will return true as soon as there is one byte available, so if your main loop is very fast you run the risk of checking for available data before all the data is recieved, reading only one byte (digit), sending that to the LCD, and start over.
You should not let the PC send to frequently, give the Arduino board time to read and process all data before sending again.
Normally it would be smart to have some kind if two way communication, leting the Arduino board signal back to the PC, "got that, i'm ready for more". In your case that would be difficult since you don't have anything on the PC to listen for that.