let’s start with a belligerent directive … if you are going to deep-etch an image, DO NOT DESTROY PIXELS. there is absolutely no good reason for permanently erasing a background. you should always use a mask instead. if you make a mistake or change your mind or whatever, it’s easier to alter a mask than it is to reclaim lost pixels. ok, you’ve been warned.
a mask is a non-destructive way to create a transparent or translucent portion of a photoshop layer. the mask appears next to the image in the layers panel. white areas are totally opaque, black areas are totally transparent, and grey areas are somewhere between :

it may be a little easier to see the difference with a red background :

different images require different approaches to deep-etching. here’s one way to do one kind of image.
we need to get rid of the white background of this image so we can put it on a black background. to start with we can make a quick selection with the magic wand tool or by using the color range command under the select menu :

this is not a perfect selection however, because the chrome highlights are white too and have also been selected. if we use this selection as our mask we’ll get big black spaces in our chrome — not pretty. refining the selection is a perfect job for quick mask mode (hit the Q key) :

zoom in and use your brush tool to paint in the missing areas. you can change the colour and transparency of the quick mask by double-clicking the quick mask tool near the bottom of your tool panel :

now you can exit quick mask (Q) and create the actual mask by hitting the add layer mask button at the bottom of the layers panel :

well, that’s not quite right — we’ve masked out the bike instead of the background. not to worry, there are at least two easy ways to fix this : either undo and hold your option key as you click the add layer mask button; or click on the layer mask and hit command-i to invert the mask.
now we can see the mask is not quite tight enough — there’s a white halo around the bike. this can be fixed by selecting the layer mask and choosing filter > other > minimum (this example only needed a 1 pixel contraction) :

the next obvious thing to fix are those shadow areas under the wheels. you can do this by painting directly onto the layer mask with your brush tool, or you could make a selection and fill it with black (again, on the mask itself, not the image) :

the only problem now is those disastrous spokes. let’s have a closer look :

now, you could try patching all that up with a lot of brush and eraser work, but here’s a simpler way. first, use the brush and eraser to completely mask out all the spokes and do a basic cleanup (tip: you don’t actually need to swap between the brush and eraser, you could just swap between a black brush and a white brush — whatever you find easier) :

next, disable the layer mask (shift-click on the layer mask) so you can see the original image. and use your pen tool to draw straight paths for each of the spokes :

here are the completed paths with the top layer turned off so you can see them easily :

make a new layer, select your brush tool and set it to a fine solid brush (this example uses a 1 pixel, 100% hardness brush), then select your path in the paths panel and choose stroke path from the dropdown menu. if the resulting lines are not quite strong enough you can just stroke the path again with the same settings — try that before opting for a bigger brush :

now select the lines by holding down the command key as you click on the layer preview in the layers panel. turn off that top layer and hit command-h (hide selection) so you can see what you’re doing. click on the layer mask and hit delete. if the result isn’t strong enough, just hit delete again :

bonza :

as mentioned, this is just one way to mask this image. feel free to share your favourite way.
keep grunting.