the phone doesn't have to run...when you connect power it generates warmth as the components start to work. In other words if something is cold it is NOT receiving any power and if it is too hot (cook an egg hot)then that part is receiving too much power and will shortly decide to go phutt (or blow a fuse and then it will be cold as the power has been disrupted) You need to find out why too much power is getting through and work out what it damaged using either the heat/cold or a multimeter to check fuses etc around cold parts.
Does your phone get recognised by itunes when you plug in USB?chime when it's on charge?It sounds like you tried to reassemble the phone when it was on? Have you taken the phone apart..let it rest for about an hour and then reassemble. Charge with wall charger for 1 hour and then try turning on/reset. If your phone was on when you assembled it will sometimes not recognise LCD etc and you get a black screen..no matter how many times you try charge/reset etc.(this doesn't mean the screen is dead it should be fine)
I think perhaps the original problem was in the dock/charging circuit but let's trouble shoot the rest first.