Specifically, most of the time Dawkins says that genes are trying to maximize their chances of "survival" (The Extended Phenotype, p. 233) but sometimes he says they are trying to increase in number (Phenotype, p. 84). When Dawkins gives examples, the genes are usually trying to increase the number of "germline" genes. (Germline genes are those actually in reproductive lineages.)
This distinction (survival vs. increase) is not a verbal quibble. The two goals produce different mathematical models and these will usually make different predictions, probably including testably different predictions. I went on to try to settle the issue of survival vs. increase, looking at the internal coherence of the two ideas, and their consistency with Dawkins methodology. Here Harshad had questions. I had decided one of the two goals was probably right, and the other wrong. Harshad wondered whether some combination of the two might be at work.
Dawkins' materialist method is the major reason for thinking survival vs. increase must be decided entirely for one goal or the other. Natural selection is driven by "selfish genes", but (from my "Dawkins #1: Are His Genes Selfish Enough?"):
Closely related to the "increase" vs. "survival" question, is the question of the exact nature of "the unit of selfishness". It's the "gene", but "the gene" can either be the collection of all genes with the same phenotype at the same place in the chromosome (which I'll call the gene-type), or one individual member of that collection (which I'll call the gene-copy). While the English language is ambiguous, Dawkins is not. His examples and his more detailed explications clearly show that he thinks the gene-type is what is "selfish", and that gene-copies subordinate their individual interests to the gene-type.If the gene-copy is the "unit of selfishness", survival is the goal. If the gene-type is the "unit of selfishness", increase is the goal. A mechanism where the two cooperate on some sort of shared goal can't be excluded, but it is hard to see how they can share responsibility in materialist terms.
The above brutally summarizes my previous Dawkins posts. I make these arguments much more carefully in them and I also examine the kind of combined goals Harshad suggests in detail.