First of all you should remove all unnecessary peripherals and cables from the Amiga, in case it is a faulty device that is preventing it from working. All you need is the power cable, display cable, and maybe a mouse. If the Amiga works in this state, then the fault is most likely to be one of the peripherals. But if it still doesn't work, then read on.
Amiga computers of all models have two particular weaknesses which often appear with age.
The first is the capacitors tend to fail as they get old. This can manifest itself as the Amiga showing some signs of life (power LED and video signal) but not booting. I've also heard that poor-quality video and audio can be a symptom of failing capacitors. Amiga models that used SMT - 600, 1200, 4000 and CD32 - will almost certainly have capacitors that are at high risk of leaking given their age (and the not particularly high quality parts that Commodore used).
The fix for this is straightforward - replace the capacitors! Those experienced at soldering can do this job themselves, and the replacement parts are relatively cheap. There are some specialist Amiga online stores who often stock capacitor packs for the different models of Amiga. If you have an SMT mainboard and you are not an expert at soldering or you do not have the correct equipment for SMT work, you would be strongly advised to find an Amiga specialist who provides a "recapping" service.
Expired capacitors can leak a corrosive fluid which will need cleaning up and may cause damage to the mainboard, so you may find that further repair work is needed. Even if the capacitors haven't leaked and/or your Amiga is working well, if you have an SMT mainboard you would be strongly advised to get the capacitors replaced anyway as they are at high risk of leaking corrosive fluid.
The second problem you're likely to see is a leaking battery. Not all Amigas have onboard batteries, but those that did used an NiCd rechargeable battery. When these get old they tend to leak, and this is more likely if it has not been actively used for a while.
A failing NiCd battery is bad news. They leak a corrosive alkaline which can eat through the traces of the mainboard, and corrode metal parts. The alkaline can cover areas away from the battery, and the corrosion can spread even further - a leaking battery can cause extensive damage. If your Amiga has an NiCd battery you would be advised to remove it immediately! Even if it hasn't leaked yet it is still a potential problem in the near future. Removing it is a simple as using snips to cut through the metal connections to the mainboard, or even just rocking the battery back-and-forth until the connections shear off. Remember that the battery has corrosive alkaline so protect your hands suitably.
If the battery has leaked, then you should immediately take steps to clean off the corrosive alkaline before the damage gets any worse. A web search for cleaning PCBs should provide some guides to this, but essentially you should use a brush to apply a weak acid (such as vinegar) to the corroded areas of the board (don't be surprised if it starts fizzing - this means that the acid is neutralising the alkaline) followed by rinsing with deionised water and leaving to dry thoroughly. Then you will need to carefully inspect and test for whatever damage the alkaline may have done. Repairing may take some advanced skills. Finally, if you still want to have a working RTC battery, then there are ways to attach a socket for a CR2032 button cell in place of the NiCd battery, which is a much safer alternative.
Fortunately not all Amigas had onboard batteries - only the "big box" Amigas and the A500+. The basic 500, 600 and 1200 do not have batteries. However RAM expansions such as the A501 may have NiCd batteries and therefore the same problem applies there too.
[Disclaimer: the above is guidance based on what I have read elsewhere and my own experiences, but it is not expert advice. Proceed with caution before making modifications to your computer.]