To build a blog to earn money you should use wordpress and look for a web host to host it on.
You can find web hosting for less than $4.00 per month and a domain name for around $12.00 per year.
When you host your own wordpress blog you have full control over it and it's more professional.
Not many blogs rank very good on blogger platform.
Once you get your blog setup then you can start writing content. Always try and post a new blog post at least once per week if possible.
A constantly updated blog get's crawled much more often by google and the more indexed pages you have in googles search results the better chance you have at gaining more website traffic.
You should also promote your blog posts through twitter and facebook.
On twitter follow twitter accounts that are related to your website. Then tweet a link to your new blog posts.
Once you have enough content and get enough website traffic then you can apply for Adsense. If approved then you can place the ad code on your blog and if someone clicks your ads then you make money.
Though keep in mind that making money blogging and with Adsense is not a get rich quick scheme.
Earning with Adsense is a long term process. And you must work hard especially in the beginning but the payoff in the future years down the road can be huge.