First off, here are the most concentrated areas.
Without a doubt, Shinjuku is the densest area in Tokyo, with its station being the busiest in the world. Inside the station and when you walk out, there are copious amounts of people walking around. Shibuya, Ikebukuro, Tokyo, and Akihabara are the next densest stations, and this mostly has to do with tourists. These all have great tourist attractions, but they aren't the best places to avoid crowds.
Generally, to avoid crowds, it's a great idea to go to more obscure stations, like stations that the Yamanote and Chuo JR lines don't go to. Even better, get off obscure subways stations. However, it isn't a good idea to get off just anywhere. I have a few recommendations.
Koenji, Asakusabashi, and Ochanomizu
Koenji is a rather less-dense area but with many places to eat and a shopping street for second-hand clothing. I often find it to less-populated here - even during the evening - not being a usual tourist destination. When it is more crowded, it's usually locals who come here. Koenji is a great place if you're thrift shopping and ballin' on a budget (check out my other article)!
(Peaceful Koenji Street. Image Source: TimeOut.com)
Asakusabashi is one station away from Akihabara - you could walk if you wanted - and there are some nice cafes and bakeries near the station for some time to relax. There are many bead and craft shops around here, which are usually never busy. The streets and stores are old-fashioned and local, giving off a more peaceful vibe.
(Enjoying a matcha latte in a peaceful Doutor in Asakusabashi. Image Source: Me)
Ochanomizu (translates literally to "tea water") is home the Guitar Street, a street with many guitars for decent prices. It is a nice place to come see and it has a great vibe. Right by the JR station is Kanda River, a great place to catch some views. It is generally never too busy here and a great place to escape the crowd, being just near Akihabara and Tokyo Station.
(Kanda River right next to Ochanomizo Station. Image Source: Flickr.com)
Other Tips for Avoiding Crowds
To avoid crowds, a good idea is to go out on a weekday morning, either right before or just after rush hour. It gets busier during lunch and again in the late afternoon and evening. Generally, nowhere is too busy early in the morning. As for times of the year, I recommend going in early summer or late autumn, where prices are cheaper and there are less tourists. As for transport, try to avoid rush hour, which are from 8 to 9am and after 5pm - just around lunchtime can get pretty crowded as well. The JR is often more crowded than other transport, so also consider taking the bus, metro, and other private railways. It also doesn't hurt to eat meals at rather unusual times (maybe because of jetlag) in order to avoid lines and crowds at restaurants as well! Visiting peaceful cafes are a great way to kill time as well - many have power outlets and free wifi!
There are many peaceful, local areas of Tokyo to avoid crowds, I recommend trying them out to relax. Remember, these are just my own personal tips, but hopefully they will come in handy!