Moondance does go somewhere. To hell in a hand-cart
This is a massive question, ML94. Walking lines are one of the ose things that take a minute to learn and a lifetime to master. I have been working on walking lines for 30 years and still struggle somedays. Really struggle. I think the simple answer is 'many things'. Firstly, impetus. You want your walking lines to add impetus to the music, to fulfil the roll of making the music swing (in all senses of the word). You need rhythmic strength but you also need to consider whether you want to add interest rhythmically. One of the things I learned from listening to 100s of hours of Paul Chambers was that, when he walks, his lines are made up almost entirely of quarter notes with very little in the way of fills and kicks along the way. Someone like Rufus Reid is much morelikely to stick in triplets, fills, double stops etc whereas Scott LaFaro etc will tear up the rhythm and play anything BUT the quarter note (not really but you get the point). In short, making the music move is an important part of walking but the way you do it is determined by [i]your[/i] musical choices and by the needs of the music you are playing.
Harmonic movement - You want your lines to spell out the chords som the listener and soloist alike can follow the form. This is, however, balanced against the need to create interest - a consistent spelling out of arpeggios is 'correct' in theoretical terms but gets boring very quickly. So, it is a case of maintaining a sense of the harmonic structure without slavishly dictating it - that is the 'art' of it, the bit that takes a lifetime to master.
As does the element of tension and release. You can use you lines to create massive tension (the most common way of doing this is playing a fourth below the root of a chord, say, on the last 8 bars of a 32-bar chorus of 'rhythm changes'). YOu can follow the soloist rhythmically, or the drummer or play against them. There are lots of ways of doing it and most of the methods are simple but, as with the harmony thing, it is the musical application of these concepts that takes a lifetime to master and which defines [i]you[/i] as a player.
You also want to create a little artwork of your own so that your lines have an internal logic and an arc that reflects your intentions. This is particularly important when playing modally. 150 bars of Dm is both the easiest thing to play and the hardest. Keep it interesting, keep it real.
It's a life's work.