What you should start with is checking the voltages that are coming out of the power supply board.If you look closely on the TV inner panel there will be a sticker on that indicates who made the panel, when it was manufactured and certain high voltages.For example you will see 165 volts and say 190 volts.The voltages are found at certain power supply and board inputs that have the same symbols.So you may also see marked on a board where a multi wire cable connector plugs into that mentions beside each pin connector voltages like 5v, 12v etc.What you need to do is ground negative lead of a digital voltage meter to one of the grounds of these pins and then test for the known different voltages at each pin.
I've used the test meter probe metal lead tip and slid it into the same matching wire connector on the back of the plug in harness as it's still plugged into the board.For example find the farthest negative pin that is hopefully a few pins over from the 12 volt etc pin you want to check.Slide the negative lead into the matching ground wire socket connector and then the positive lead into the 12 volt wire connector.Then turn the TV on via it's remote or manually and see if 12 volts is at that pin.Note you should test the pins too at the board to make sure that the proper voltages exist without the wire harness plugged in.This can be done with some alligator clips and then make sure you slide the alligator clip rubber down far as you can over the pin its clipped to.Do the same with the other ends of the test alligator clips and the meters metal lead ends.
After you know that the board has all the correct voltages at the pins and when the harness is plugged in, try testing the other end of that harness when it's plugged into whatever board it feeds.This will tell you if the power supply board has the proper known voltages at those pins and also that the wire harness seems to be working fine.Then pop that board out that it plugs into via that harness and check each soldering connection where the pins on it's board connect up to.Moister/dampness from storage damages etc could cause bad or corroded or broken or faulty soldering connections.Note removing the old solder, pin by pin and adding fresh solder might be the best bet and before re-soldering clean the soldering surfaces/points.
You tried replacing T-Con Board and that didn't fix it so you need to know if the power supply feed to that board are all okay.Note that the boards may not just have the standard fuses some boards have tiny mini fuses that can be spotted by the fuse symbol or it will say for example F101 on the board by the fuses.
Boards like the tuner board could cause the no video issue and since you mention something about a weak or faint signal of the different channels it could be the tuner/main board.I'd want to test/check the coax cable input to the main board. Then test the cable,AVI, DVI , HDMI ports to make sure that it's not just one input signal that isn't working properly.
I do know that Flat Screen TV's that are 3-5 yrs or older are known for bad connections and bad capacitors.So bec it's a LCD TV no video issues like that have to be either the main board or the T-con board.Especially if the black lights seem to be working fine then you know bec a supposedly good T-Con was installed that either a bad power feed exists or the Tuner board or main board as many call it has to be acting up.Again you have to make sure no bad connections exist and work from there.
Hope that helps and best of luck.By the way you should check all the panel ribbon wire connections to make sure that a ribbon has rotted/corroded bec of the moister etc.If that is the case the TV is good for junk but the boards might come in handy.Personally if I knew that the panel was fine and the ribbons are still connected solidly I'd try to locate a as is TV just like yours and use it's parts to fix yours or fix it with your parts.It can be hard to tell if a capacitor is gone or faulty so what I'd suggest is if you really find that the voltages are wrong coming from the power supply board replace or fix it. Replace any of the capacitors with low ESR rated capacitors .Replace all the capacitors that are especially mounted close to sources of heat like close by a Transistors or heatsink etc. Some of the main boards and power supply boards etc can be expensive to replace but I've seen them go for as little as 35 bucks or up to well over 150 bucks on ebay.Note try to use the board numbers and panel numbers plus the TV's exact model numbers to try and match boards or a as is parts TV on ebay etc.