First, oil should not be put down the sink, because it can lend itself to blockage in the drain.
Oil should be bottled when no longer needed, or stored for further use. You are asking how to strain out the crumbs and stuff, as you bottle up the oil for re-use. You would not care, if you intended to discard the oil. You should always smell the oil and decide whether it is worth saving.
My technique is to let the oil cool over night and allow the crumbs to settle. Then I pour off the top oil which does not need to be strained. A small hand strainer that can be held over a funnel inserted into the bottle neck is the best strategy.
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The small one on the right in this photo, is the one I use for the purpose. Coffee and tea strainers won't fit into that, but you can cut a piece off one and settle it in the strainer, before starting. Personally, I think going to coffee strainers slows things down and I would not go that route. I usually just strain the remaining bottom oil off into a taller container, let it sit again, till the debris settles and then pour off the top oil again, perhaps repeating this process a couple times, finally discarding the remaining dirty oil in a throw-away container that can be tightly sealed. Going after the top oil in successive passes does not put you in the position of waiting on a coffee filter to strain a little bit of oil that will fit in the strainer while you holding the strainer and getting tired of doing so. Going after the top oil does have delays to wait for successive passes to settle out in taller containers, like tall narrow jars, but those delays are not great, if you take care.
Going after top oil is best in my book. I do strain the oil in any case as I pour the top oil off into the storage container.