Can I get my contacts back from my dead iPhone 5s?

Try a hard reboot, just in case:

Press and hold Home Button and Power Button together for ~12 seconds.

Then release Home Button, but keep pressing Power Button until iPad restarts (1~3 seconds, in case it works).

EDIT:

In the case of iPhones, they have just one main board, and the data is contained on certain chips that are soldered to that board.

To recover your data, if the iPhone is "fried" indeed, you'll need a professional repair tech trouble shoot and repair it. It could be that some parts of the charging circuits are damaged, such as some caps, resistors, U2 IC, and those are quite replaceable by the right person using the right combination of skills, experience, and tools. It could be that the damage extended to the Power Management IC, or worse, to the CPU. In which cases it becomes more and more difficult to repair and the data you have should be really worth the trouble and potential expenses.

However, before jumping to conclusions, you can see if by any luck only your charging dock is damaged. You can either have it examined by a reputed repair shop (or sent out to someone), or try to replace that charging dock yourself using this guide that features all the needed steps, tools, and parts. In the process, you'll be disconnecting the battery and thus power-cycling the phone, which could also help in case it is just "stunned."

Just make sure when you re-assemble the phone to follow the instructions to the tee, putting back every screw exactly where it came from, and make sure to test/charge the phone using a known good cable and a known good wall charger or computer USB port.

EDIT #2: after going to the repair technician:

You are lucky - somewhat.

When you charged from the train, you apparently damaged the U2 IC.

Then the U2 is damaged, 3 things begin to happen:

1/ Phone discharges extremely quickly. Sometimes even when charging, the % barely increases.

2/ Once battery is empty, the phone cannot recharge it again. The phone will "say" it is charging, but will not. To restart the phone, you have have to install a battery that is already charged.

3/ Soon the phone cannot communicate with the computer via USB cable.

Why is the technician able to start it? He's possibly putting a charged battery in. Once it gets to 0% though, the phone won't be able to recharge it. This is why he said to keep it charged at all cost.

Is this repairable? Yes it is, but not many technicians are able to do it.

Is there a risk? Yes, during the repair, you may kill the board.

Can you replace the board? Yes, but there are associated risks such as locked to iCloud, suffering from hidden damage. And you have to find one with a matching home button if you insist on having a working Touch ID function

My email is on my profile. Send me an email with your location and I'll recommend someone based on that. Some contributors here can perform this repair and you can indeed send them the phone by mail.