Is there a problem with citing an article that cites a retracted article? I'm looking through a few journal articles and I notice that they have cited an article that has been retracted (based on invalid conclusions).

If I cite that article that cited the retracted one, would that be valid? Moreover, should I check the references of every article in that article to see if they cite any retracted articles?
Ultimately, it doesn't depend on the retracted article, but on the one that you cite.
Is the science in the article that you cite valid, despite the retraction of its reference?  After all, most references are for context not critical dependencies. If it's still valid, there is no problem with the citation.  If the science doesn't stand, though, then it doesn't stand and you shouldn't cite it.