So this question is difficult to answer because it has some errors in it, in that lifeforms being dependent on rarely used elements is a different question than lifeforms using different DNA bases, but I will answer both.
So for the question about whether or not organisms could exist that use DNA bases not found in organisms on earth, the answer is certainly yes.[1,2]
The above reference shows a very elegant example of completely synthetic DNA bases being incorporated into E Coli and propagated in culture without providing any noticeable negative effects. This provides strong support that life on Earth developed around the same 5 bases (CTGAU) by chance.
As for whether or not organisms could rely heavily on elements other than H,C,N,O, and P, this is certainly possible and evident already here on Earth. Extremophiles are a general class of organisms that have some pretty interesting properties to them. Some metabolize sulfur or metals, others live in really extreme temperatures or in areas of high radiation.
So really yes in both cases, life can really take numerous shapes and forms even when it seems completely impossible for them to do so.