Just get an old clean cloth for oil and apply the oil down the woodgrain.

Note that lemon oil is for rosewood and ebony only.
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Don't use any old cloth. Normal cloth and tissue can (will) leave fibres along the fretboard. These can get under the fretwire, in the grain and in the nut slots, causing horrible tuning problems and you'll never be able to spot what is causing it. When cleaning any part of your guitar, you shoudl only ever use dry, inkless microfibre cloths. You can buy these at most opticians and photography specialist stores; some guitar stores may also carry them, though these will probably be more expensive than ones you would buy at an opticians (and won't work any better). These are the one and only type of cloth or applicator you should ever use on your guitar.

For lemon oil, you need to check just how dried out your fretboard is. If it is only in the beginning stages of drying out then you shouldn't need any more than two to three drops/quick squirts of the lemon oil on to the microfibre cloth. Give it a few seconds to sink in to the cloth (you don't want it just sitting on the surface). Now lightly rub the cloth over the exposed and most dry areas of the fretboard, using smooth circular motions. Be careful not to go near the nut, though you can risk going near the fretwire. One pass over the fretboard like this should be enough.
If your fretboard is completely dried out, do the above but repeat it a second time. If your fretboard is in really dire condition you may need to give it a third very light pass, but if you do it a third time make sure to stay clear of the fretwire that time too, as well as the nut.

Obviously, do not get any oil on the finished parts of the fretboard (usually the sides and back).

Bear in mind that there's a good chance your fretboard does not need oiling at all. A decent quality rosewood fretboard that has been stored away from any particular heat sources, has not been left in direct sunlight for months on end and has been played on a regular basis won't need oiling any more than once a year at the very most (and that is if you have particularly dry skin). The average rosewood fretboard, kept in average conditions and played on a regular basis by someone with average to slightly oily skin, can go without extra oiling for easily two years. For reference, I've got a guitar with a rosewood fretboard here that hasn't needed oiling for almost five years.

With ebony fretboards, repeat all of the above only they need oiling even less than rosewood fretboards do. We're seriously talking once every five years even for an average board when it comes to ebony.
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