The simplest answer to your question is that ATR can phosphorylate Chk1 on Serine 345 in response to various DNA damage-inducing insults (e.g. hydroxyurea, UV etc.); this phosphorylation promotes Chk1's own kinase activity and allows it to act on its own targets (e.g. Cdc25C, p53.....).
However, it's worth remembering that DNA damage repair signalling is a very complicated pathway, and many things can activate many other things - it's better to think of it as a network than as a pathway! So, although ATR can act on Chk1, this action can be altered by other signalling cascades or may depend on the type of DNA damage induced, cell type and many other criteria.
The first paper to describe ATR-Chk1 regulation was Liu et al. (2000), doi: 10.1101/gad.14.12.1448.
For a recent review, see for example Smits and Gillespie (2015), doi: 10.1111/febs.13387.
RPA-covered single stranded DNA, such as arising from replication fork stalling or genotoxic insults (e.g. following DNA double strand break end resection), will lead to ATR activation. How extensive the subsequent phosphorylation of Chk1 is depends on various factors, including the presence of claspin etc.