Have you taken the inner tube out after you've had the issues inflating it? was the tube still intact after taking it out? (no leaks)? If there was a leak there are a few things to check:
make sure the rim tape is in good condition (no tears/holes, make sure it's properly aligned in the rim)
remove rim tape and check if any spokes protrude more than approx 1-2mm past the top of the spoke nipples, if they do consider shortening these spokes with a file, Dremel or the like or adjusting the spokes such that they are no longer protruding (if possible whilst keeping the wheel true/round)
check rim bed/edges (where the tire contacts the rim) for damage (sharp/pointy spots which could potentially damage the tire), if you find any sand/file them down so they're smooth
check your tire for pieces of metal/glass/other objects embedded in/under the surface (by rotating the tire whilst removed from the wheel and pinching it every 2cm approx to reveal any splinters, do this on both the inside and outside of the tire
check if valve hole has any sharp edges
whilst installing inner tube make sure it is not pinched in between rim and outer tube (inflate inner tube just a bit, check if it is not pinched anywhere and if all is good inflate the rest of the way)
For fixing flats I personally find it useful to clamp the inner tube with patch in between a bench vice (potentially with some pieces of wood in between the vice and the tube/patch for wider inner tubes), you could also use screw clamps for this purpose.
How far is your valve stem protruding through the rim? Would it be enough length to let the pump properly 'grab' it without leaking? If the valve stick-out is very minimal you will have to get an inner tube with a longer valve stem to be able to properly inflate it. There are also valve extenders for Schrader (and other) valve types, which you could consider using if your current valve is too short for the rim. ( (https://www.bikesale.com/slime1-14schradervalveextenders4-pack.aspx)
Edit: Could it be that when trying to install the pump onto the valve you're pushing the Shrader valve stem into the rim (it can easily be pushed down so it sticks out less when the tire is not or barely inflated). What I often do to try and prevent this is to use one hand to install the pump and use the other hand to depress the outer tire at the position of the valve stem, in this way i am preventing the valve stem from moving down when exerting pressure on it by pushing the pump onto it. If you have a pump which screws onto the Shrader's valve stem this should be a non issue, if you have a pump which 'clamps' onto the valve stem you might want to try this.