There is no time that is best to clean the gutters on the eaves. They should probably be cleaned early before the debris within them gets dampened by the first rains. However, they will only need to be cleaned out again after more foliage falls. If cleaned only after all the foliage falls, they will be grungier, and there will be potential for some of the debris to flow into and clog the downspouts.
Most of the fresh leafy debris that fell recently is relatively easy to clean out. Debris that has been accumulating through the year will be more decomposed and settled in. Evergreen trees are somehow messier than deciduous trees. They drop smaller volumes of debris in autumn, but they drop the rest throughout the year. Deciduous trees drop all their foliage within a limited season.
That certainly does not mean that deciduous trees can not make a mess. All that foliage has to go somewhere. If the weather gets cool slowly, foliage falls slowly, and for a longer time. Some deciduous trees innately defoliate slowly, and may even wait all winter to finish. Fruitless mulberry, tulip tree and poplars typically defoliate efficiently, making a big mess that gets cleaned up once.
Flat roofs that lack gutters collect debris too. So do the spaces behind chimneys and in roof valleys (where the slope changes direction). Even if this debris does not interfere with the function of gutters and downspouts, it promotes rot in roofing material. Trees and vines that touch a roof are likely to be abrasive to roofing material if they move in the breeze, or hold debris against the roof.
Trees and vines must be kept clear of chimneys, not only because they can interfere with ventilation, but also because they can be cooked by exhaust from the fireplace below, and ignite! Fan palm beards (dead foliage that accumulates on trunks), pine, cypress, spruce and cedar are particularly combustible. Clinging vines can separate brick from mortar, which is another fire hazard.
Working on the roof and gutters is of course potentially dangerous. It might be best to get a professional for these sorts of jobs, especially if trees need to be pruned. While that is being done, there is plenty of raking and other gardening for us to do (without a ladder). Raking leaves is an important job too, since fallen leaves can shade out lawn, ground-cover and flowering annuals.