Tokyo and Japan on the whole have four distinct seasons and the impression you'll get can be completely different depending on when you arrive.
Top choice: May or October for the most stable weather and pleasant temperatures.
Also keep in mind: Winters (Dec-Feb) aren't as bad as you think - with dry weather and the best shot of seeing Mt Fuji.
When the plum trees start to blossom in late February or early March, it's the first sign that spring has arrived. The weather can be unstable in early spring, as there is an old saying that season has "three cold days and four warm ones"
By late March, the much awaited cherry blossom season arrives but when you can see it in Tokyo can vary depending on the year. In 2023, the blossoms came out extremely early and were mostly gone by the end of March. However there were some late blooming ones remaining until about April 10th if you searched carefully. The usual schedule would see the peak in the first week of April and this is when you are guaranteed to see the blossoms somewhere for sure, though this might mean having to go a little north or climb to higher elevations find one.
In April, it's normal for the highs to climb above 20C(68F) making for a pleasant day outside. However you may want to avoid travel around Apr 29 - May 5 which is the long holiday period locally called "Golden Week". Virtually the entire nation travels during this period when planes and trains become packed. Hotel prices will also skyrocket and you may be standing in lines for hours to get to your destined attraction. Car travel should be avoided - a drive to the Mt Fuji area from Tokyo can take 5-6 hours for example, double or triple the normal times! Things relax considerably after the Golden Week is over so if you can avoid this period, the spring is no doubt the best season to travel.
Great weather continues until early June when the rain season called Tsuyu arrives. It can rain the intermittently for few days straight and the humidity can stay above 90% making for unpleasant conditions. By early to mid July, just when the schoolkids enter summer break, the Tsuyu is finally over and summer is officially here. While there are many festivals and fireworks shows in the summer, you may find that the climate is similar to that of a tropical area such as Singapore or Bangkok. The temperature sometimes reaches 35c (95F) and remains above 25c (77F) even in the nighttime. You may want to plan some indoor activities such as shopping or a visit to the museums during the daytime to avoid the heat.
In September, things finally start to cool down a bit, and conditions should be preferable for sightseeing. However you do need to watch out for typhoons that occasionally make landfall during this time of the year although the storms usually pass through quickly. In recent years the rail operators tend to halt operations entirely when a storm is approaching. The announcement will be made 1-2 days before, so look out for notices. Communicate with hotel or tour operator staff if you see a storm approaching and seek information if you have doubts. The typhoon season continues through early November.
October will be the peak season for international tourism as many cruise ships call the Japanese ports during this month. The weather is generally stable but towards the end of the month, the low may dip below 10℃ (50F) so you will need to be careful about what type of clothing your bring.
By November, the leaves will start to turn colors in the areas up north such as Nikko and Hakone but it is usually not until the first week of December when you conditions reach the peak at locations in central Tokyo.
Tokyo has a fairly mild winter, at least compared with many major cities in the northern hemisphere. In the afternoon, temperatures may rise to around 10℃ while the lows rarely dip below the freezing mark. Skiing can be enjoyed in the mountains of Nagano or Niigata about 2 hours north, but snowfall in the city is extremely rare. In Feb 2014, 27cm of snow fell in Tokyo which closed the highways for several days but this is a rare exception. If snow is forecasted in central Tokyo (even just a few centimeters), the rail operators tend to play it safe and may reduce or shut down services.
December - February is the driest months of the year and it is not unusual to go weeks without any precipitation. Visibility is the best in winter, meaning chances of seeing Mt Fuji from Tokyo is very high.
Be aware that many museums close for the Year-End/New Years, from around Dec 28 to Jan 4, although shopping malls and restaurants generally stay open. Trains continue to run as normal. Many workers in Tokyo return to their hometowns for the holidays so the long distance trains and flights can become fully booked.