Most HP cartage can be refilled from the top, I.e ink is put in via needles from the top of the cartage.
The ink is held by a sponge like substance which has to be poked in by the needle and refilled. Poke deep into the sponge, but not so deep so as to touch the base. A bit below the middle most depth will be good.
First, before you start to refill, seal off the printing heads (the steel part) with cello tape to avoid touching.
For black cartridge, open the top sticker, you'll see 5 holes. Refill using the middle most (and largest) hole. Refill till you see a little bit of liquid ink on one of the 5 holes. If you filled in excess, remove it using the suction tool. Sign of excess is that it starts dripping from the other end (nozzle). If you fill in excess, you'll waist ink and ruin the cartridge.
During refilling, do it slowest possible and wait at times between pushing the ink (do it in sessions). If you do it fast, it'll look as if the ink has been filled (the symptoms will appears), but in reality it's just a shallow fill.
Seal off the exposed holes with tape.
Wait for 24 hours before starting to print. Put the cartridge in the horizontal position (printing heads facing the floor). This's to allow the ink seep into the printing heads. If you do it earlier, you may see faded out prints; which must not be mistaken for low ink. You just need to wait in case that happens to a newly filled refill.
Create a test page to print black. If it does not, use the suction tool to open up the blocked pores. Do minimal amount of suction so as to just bring ink to the syringe. As that happens, the pores will get opened up again.
For color cartridge, you'll also see 5 holes, the largest hole is for Magenta, left for yellow, and right for Cyan. This is when the cartridge's nozzle (or the chip) is opposite of you or the Magenta hole is on the top.
For 703 cartridge, the scheme is a little different. First, all the holes are of the same size. The top most (TH) is yellow (with the chip NOT facing against you and the middle most (yellow) hole far away from you). Left is Magenta and right is Cyan.
In the left and right, there will be a pair of holes (PH), you can use any of the 2, but it's recommended to use only 1, I'll tell you why.
When you see the holes in light, you'll see them having a sponge, and on the sponge there'll be ink stains. For PH, use the hole with the ink stains.
To do the actual refill, you need to follow similar protocols to that of the black refill, with a little bit of modification.
So you did realize this's a sponge right? You need to refill till the sponge gets filled up with ink; i.e. it gets all wet, but don't fill in excess, it must get JUST wet; the wetness must be such that the sponge starts looking almost completely black and wet. As that happens it may happen that the other hole in the PH starts getting wet too, if it does stop refilling.
For the tests, use Cyan, magenta and yellow colors separately.