In an audio circuit, with an LM386 chip - how does the DC electric power and the AC audio work?
The output stage of the LM386 can be simplified to two transistors (NPN &amp; PNP) for examining current flow. I have re-arranged output to show R1, a 10-ohm resistor in series with the 250 uf capacitor C1. Since they're in series, it doesn't matter which goes to ground from an electrical point-of-view. The usual safer practice grounds the speaker rather than the capacitor. For this discussion, it may be easier to see how the capacitor gets charged, if one of its ends is ground-referenced....

C1 must start off discharged. Both its ground end and its "hot" end (node 0004 in the schematic) are at zero volts initially at t=0. But eventually, C1 must charge up to about half the supply voltage: this is the bias condition imposed by the internal structure of the LM386: in this case it will charge to +5v. When you first apply power, you may hear a "pop" from the speaker - this will be the charge current that gets C1 up to +5v.
Your question involves the simultaneous AC and DC currents that flow. So I've included an AC source (V2) that drives a sine wave into Q1 &amp; Q2. In the SPICE transient analysis, this 1 KHz sine wave is superimposed onto the charge current for C1. C1's voltage is plotted (red trace). The AC voltage amplitude is small compared to the AC voltage amplitude across R1:

Q1 must initially work hard (green trace), because it is burdened with both charge current for C1, as well as supplying the AC load current for R1. Q2 is lightly burdened initially (dark blue trace). Of course, nearly all the charge current is supplied by V1, the DC supply.

Note that current only flows one way from V1 supply (through Q1). When Q1 is off, Q2 drags current in the opposite direction, using C1 as the source of current.

Once C1 is finally charged to +5V, Q1 and Q2 are equally burdened with current flow. Since these currents dominate current flowing into LM386 pin 6, you could monitor current flow there - it would appear similar to the green trace.
Since the current pulled from the DC supply varies from near zero to a large value, it is a good idea to add a big bypass capacitor from pin 6 to ground to smooth any supply variations.