You can put your site's link on websites that you design and build up a portfolio by looking at sites like Craigslist or whatever to see who is looking to get websites developed. You likely should spend more time worrying about SEO than backlinks to your site..
You actually don't want to put a link to your site on just any site that you design. Google needs to see that there is some relevance between the content on each site. So if you help to design a fashion website, let's say, and put a link on it to your web design website, then Google won't see the relevant connection and actually ding you on it.
Ultimately, you need to first create high-quality content like DrQuincy suggested. Content should be keyword optimized too in order for your site to rank higher in the search engine. SEO is actually not a waste of time or money. You can learn a lot of the basics on your own for free in fact, and spending the time to implement this stuff will help you see a big difference in your website performance. As your website performs better in the search engine, you will begin to accumulate some relevant backlinks. There are a lot of ways for you to actively build backlinks. All you have to do is Google that in fact and you will come up with hundreds of articles telling you what to try,
like this one, but none of this is going to be quick and easy. It is a major time and effort investment. So do your research and through trial and error, see what works for you.
JUST DO NOT buy links or stuff links in randomly. Ultimately the rule is, if it seems too easy, it is too good to be true. Google knows all, and it will see through that right away as a ploy to just gain links and that type of activity will end up getting your site banned.
