First, check out and install the depot_tools package.

$ git clone https://chromium.googlesource.com/chromium/tools/depot_tools.git
$ export PATH=$PATH:/path/to/depot_tools

Checkout source code

$ mkdir ~/chromium_build && cd $_
~/chromium_build$ fetch --nohooks android 
//Configure GYP
~/chromium_build$ echo "{ 'GYP_DEFINES': 'OS=android', }" > chromium.gyp_env
//Update projects from gyp files. 
//You may need to run this again when you have added new files, updated gyp files, or sync'ed your repository.
~/chromium_build$ gclient runhooks

Environment setup

~/chromium_build$ sudo apt-get install openjdk-7-jdk
//config default JDK
~/chromium_build$ sudo update-alternatives --config javac
~/chromium_build$ sudo update-alternatives --config java
~/chromium_build$ sudo update-alternatives --config javaws
~/chromium_build$ sudo update-alternatives --config javap
~/chromium_build$ sudo update-alternatives --config jar
~/chromium_build$ sudo update-alternatives --config jarsigner
//install build dependencies
~/chromium_build$ src/build/install-build-deps-android.sh 
// Install Google Play Services
~/chromium_build$ src/third_party/android_tools/sdk/tools/android update sdk --no-ui --filter 57

Build

~/chromium_build$ cd ~/chromium_build/src
//Full browser
~/chromium_build/src$ ninja -C out/Debug chrome_public_apk
//WebView
~/chromium_build/src$ ninja -C out/Debug android_webview_apk
//sync source
~/chromium_build/src$ gclient sync

Android APK location: ~/chromium_source/src/out/Debug/apks/

Reference:
Android Build Instructions
depot_tools_tutorial Manual Page
Getting Around the Chromium Source Code Directory Structure
Organization of code for Android WebView
Design Documents
HW Video Acceleration