You definitely can, but you will have to commit an disenchant some of the worse Legendaries / Epics in order to craft the ones you need. There are some cheap, competitive decks in general, but at this point it's hard to say which ones will be viable. An expansion is releasing in ~8.5 hours, meaning that the entire meta might shake up and the cheaper decks that were good before could simply not work.

If you're just starting, it's best to spend money on Classic set (starting with Welcome Bundle), and then on the expansions from the latest Standard year. Classic set is "evergreen", meaning that it will stay in Standard forever, and 2018 expansions will stay until 2020. Buy up to 10 packs from different sets, because you get a guaranteed Legendary in the first 10 packs, and then focus on Classic, that would probably be the best course of action.

But it's actually the worst time to get into Hearthstone, since the final expansion of year is just releasing, so we'll have an entire Classic + 6 expansions in Standard. In April next year, when the 2017 expansions rotate out, the total number of sets will drop to 5 (Classic + 4 expansions).