What are the best methods to have good quality backlinks? How long its going to take for one person to get backlinks quickly?

Nope there are no shortcuts in getting a backlink, and one more thing just getting a backlink doesn't make sure that you are going to get a bulk of traffic that totally depends upon like is there any use of putting the backlink over their ? Like if you are pushing you link without any use then I don't think its going to get you traffic rather than those link will start acting as poision link.

IMHO, there is no best method. Having good quality backlink requires more patience and time. You can make blog post according to your niche, but it should have rich content. It should be written for the purpose of giving useful information to people and not for building link.

you offer web services, and are ignorant of required information to properly design a website to minimum standards.
Go away and read google 101,
it is information that should be the basis of your business
the sites you build will otherwise be as poorly ranked as your own, and no return business, no new business.
Incidentally posting a link to your site on this thread, that says "I don't know what I am doing" DOES NOT help your business

What's important with backlinks is the quality of your content and the relevancy of the website you put your link in. Google likes content that is fresh, unique and relevant. It is also as important that you post your link in websites that are relevant to yours.

Just perform the usual link building activities in procedural way without over promotions.The backlinks would be generated only after a certain period of time.Instant backlinks are not possible.

Write a quality article write up for creating backlinks and social bookmarking by using rich content.

What Bob said. What is it with the number of folk who seem to be setting up as web designers/developers/seo specialists who know absolutely sweet FA about the subject?

Seriously, if you guys cannot even grasp the concept of Googling to get the information without posting to an industry forum where everyone can laugh at your ignorance and sheer brass balls then, frankly, the only place your business going is south and fast...