Would it be accurate to say that according to Hume causality is another way to describe the (relative) permanence of steps in causal chains of events? If this is not the case, what would be considered a brief and accurate portrayal of causality, according to empiricists in general and Hume in particular.
▻ THE BACKGROUND PICTURE
Hume is an empiricist - let's state the obvious. This means in his case that the basis of all knowledge is perception - not just visual perception but sense-based experience. You cannot have any concept or belief (idea) unless you have had a preceding sense experience (impression). Given impressions, our ideas can take indefinitely many varieties of forms. I can form the concept of a centaur, for instance, but only because I have previously had an impression (a perception) of a man and a horse. What I cannot do, Hume believes, is to form a belief or a concept that has no link, however, tenuous with sense-based experience. (A Treatise of Human Nature, I.1.1.)
▻ THE CONCEPT OF CAUSE
When he considers what he takes to be the standard or prevailing concept of causation he thinks of it as involving a necessary connexion between cause and effect. Causes necessitate their effects; given the cause the effect cannot but occur. This presents him with an immediate problem, namely that we cannot in his view perceive necessity. I can see the kettle; I can see the flame beneath it; I can see or feel the water heating and eventually boiling. I perceive this sequence of events but I cannot perceive the necessity for the kettle to boil; I perceive only that it does boil in these circumstances.
☛ Hume's alternative
Hume's alternative view of causation is that we apply the concept of causation when three conditions apply :
A causes B (we say and think) when (i) A is prior to B (occurs before B); (ii) A and B are 'contiguous' (close) in space and time, and (iii) A-type events are regularly, in a 'constant conjunction', followed by B-type events.
In his own language, 'WE may define a CAUSE to be "An object precedent and contiguous to another, and where all the objects resembling the former are placed in like relations of precedency and contiguity to those objects, that resemble the latter." (A Treatise of Human Nature, I.3.14.) He does offer immediately afterwards a second definition : 'A CAUSE is an object precedent and contiguous to another, and so united with it, that the idea of the one determines the mibd to form the idea of the other, and the impression of the oe to form a more lively idea of the other.'
The first definition is the one Hume mainly works with; also the second seems to involve circularity since 'determines' suggests causation.
▻ A PROBLEM FOR HUME'S ANALYSIS
The problem that faces Hume, or one of the problems, is that if the standard or prevailing concept of causation involves the belief in a necessary connexion between cause and effect, this idea should be impossible to form on Hume's account of impressions and ideas. We have no impression - experience - of necessary connexion, and if all ideas derive from impressions there is no impression of necessity for the idea of necessary connexion to derive from. Yet we have, Hume admits, this very idea.
He does not deal satisfactorily with the problem, which is in fact an insuperable difficulty for him. He handles it by suggesting that the constant conjunction of A-type events and B-type events makes a 'customary transition' (Treatise, I.3.14) from the idea of one to the other so smooth and automatic that it seems that a necessary connexion is involved. This is an illusion of necessary connexion but he still has to explain, which he can't, where the idea of necessary connexion comes from. How can we entertain the belief in a necessary connexion, even if it is an illusory one, if we have never had an impression, or perception, of necessity ?
▻ OTHER EMPIRICISTS
You refer to 'empiricists in general'. I am not sure that the views about causation of empiricists in general can really be stated; there is so much variation as well as dispute about who counts as an empiricist. If you have in mind Locke and Berkeley, there is a chapter on causation in Locke's 'Essay concerning Human Understanding', II.26. The key word is 'power'. Berkeley's ideas about causation are principally given in his Philosophical Commentaries but there is material in the second of the Three Dialogues between Hylas and Philonous.
Hume, A Treatise of Human Nature, I.iii.1-14; IV, 7.
Donald C. Ainslee, Hume's True Scepticism, Oxford : OUP, 2015.
G. Dicker, Hume's Epistemology and Metaphysics, London : Routledge, 1998,ch. 3-5.
D. Applebaum, The Vision of David Hume, Rockport, MA : Element, 1996 ('The Illusion of Causation', 21-4).