How should I play two notes together on lines below the staff?
Ah, that is actually two notes: 'A' and 'C'. Whenever there are notes below the normal staff, you add those lines to notate which notes they are. The notes above and below the staff are called ledger lines. If you want to figure out the notes that are below and above the staff, just cycle through the musical alphabet (A B C D E F G).
If you are trying to figure out the notes above the staff, count through the alphabet alternating from space to line (or line space) and on until you reach your note.
If you are trying to figure out a note below the staff, count the alphabet backwards. You still need to alternate from line to space while cycling through the musical alphabet. 1
there is a middle c on top of a middle c.
I think you may be confusing the general term ledger lines with the specific middle C.
The basic staves are 5 lines.
The staves take a clef to set what letters apply to the lines and spaces. Also, scientific pitch notation adds an octave number to letters so we know exactly in which octave a tone is. The treble clef is the "G" clef setting
G4 on the 2nd line and the bass clef is an "F" clef setting
F3 on the 4th line.
It may seem odd, but clefs can be put on other lines and that will change the lettering of the staff.
Ledger lines are added to a staff to go higher or lower than the 5 lines. Ledger lines can be stacked above or below each other to go higher and higher or lower and lower to get to whatever pitch needs to be notated.
Another thing that can be done to notate a note that goes off of the five lines staff is to change the clef. But, clef changes can be hard to read if you don't practice a lot of reading so people try to avoid them. The cello part below shows a clef change used to avoid going 5 ledger lines above a bass staff.
Finally, middle C is
C4 and depending on the clefs used it can appear on various lines of spaces. Here is middle C on various clefs.
So, put that all together for your example (and add the missing clef for clarity...)
...rhythmically you will strike the two notes at the same time on the piano, probably using your thumb and middle finger, but I don't think the exact fingering is part of your question.
After my mild diatribe - related mainly to the fact that there were't two middle C notes, I'll attempt to make amends by answering the question.
Imagine the standard grand staff, with its ten lines, split, as it is, into treble and bass clef (normally). That's how it's been for many a long century. Think about the fact that actually, middle C does not have a home. There is no permanent place to put it.
It's always had to have its own special little line - as it does go on a line - and that line is a temporary one, which goes under the treble clef. It also goes above the bass clef. It can be considered as the missing line out of a possible eleven, slap bang in the middle. Hence, middle C.
If there were eleven lines, it would be very difficult to pinpoint dots, so it's much simpler to have the staves split. Hence the two parts and a floating leger line between.
Now, we come to the crux.
Sometimes, there needs to be notes lower than middle C, written 'in' the treble clef only - as in the OP's example. That top note is obviously middle C, as pointed out. The note under it is effectively written on the top line of the bass clef, which obviously in this music is missing! So there's middle C (top note), and on that next line down there's note A. That's what it would be if it was on the non-existent top line, bass clef. As other answers point out, those leger lines can be expanded and sometimes several are used - both above or below either clef. Often it's the simplest way to portray notes whose home isn't within a stave.