As with any painting project, the first step is to ensure that you have a clean surface to work with. Aluminum gutters certainly can be power washed, but it's unlikely that power washing alone will remove all the accumulated grime and chalky residue that tends to build up on them. That said, it will be necessary to climb a ladder and use a brush and soap to do a detailed scrubbing by hand. Regular dish soap can be used, although automatic dishwasher detergents are more effective on gutters that may have accumulated grease or other stains from kitchen vents. Now is also the time to scrape away any loose or flaking paint with a wire brush or steel wool. Let the gutters dry completely and choose a primer.
Use a quality primer on your gutters. There are two different types from which to choose: oil based metal primers are commonly used, or you can work with an acrylic bonding primer. The acrylic primer adheres well to prefinished and glossy surfaces as well as gutters that may be worn and are still shedding some chalky residue. It is important to match primers with compatible paints, so if you do choose an acrylic primer make sure that the paint you pick for your final top coat is acrylic as well. The same applies to oil based primers; choose a quality name brand such as Behr, Dutch Boy or Benjamin Moore from any full service hardware store or home center, and match it with a compatible oil based paint.
Apply the primer with a brush and let dry as per the manufacturer's instructions. It is also an option to use a sprayer, but the time and effort spent masking the areas around the gutters may be prohibitive. Apply the topcoat of paint with a brush as well and let dry again. To ensure the best looking result it's important to sand any rough spots or visible brushstrokes with steel wool or 240 grit sandpaper. Once smooth, retouch as necessary and your gutters will look as good as new.