There are published many weather display projects. Most use an Arduino for the sketch to run on.
These Arduino based projects often use the spartan AT-command set.
This project uses the ESP8266WiFi.h library with the Arduino-ESP8266 IDE add-on
The component count is really very small. Just a ESP8266 and a small Oled display. A few resistors and a capacitor.
Ofcourse you need a 3.3V powersupply or 3 Ni-Mh batteries. Probably a Li-ion battery of 3.6V will be good too. Because a fully charged Li-ion cell will be around 4V a simple diode can be put in series with the cell to bring voltage down to less than 3.6V.
Code can be found on: https://github.com/costonisp/YAW-ESP8266/