The main problem with driving P-channel high-side switches from logic levels is that the gate voltage needs to go to the source voltage level for the switch to turn off. That's 60 Volts in your situation, and the output of a logic control is only 5V.
The simplest way to fix this is to use a small N-channel signal transistor to pull the P-channel gate to ground, and use a pretty small pull-up to pull the gate voltage to source when that N-channel is off. If you just switch on/off occasionally, a 500 Ohm or 1 kOhm pull-up may be enough. If you're doing PWM, you want 100 Ohms or less to make sure the switch happens fast enough to not blow the device.
The problem with a small resistance for pull-up is that it "wastes" current while the P-channel is on, as the N-channel signal device will keep current flowing.
Additionally, in your case, because it's a 60V potential, you have to protect the gate/source voltage differential; many devices will die if that goes below -20 V or so. Thus, you want a small resistor on the pull-down side, too, and a Zener diode on the high side to limit the max voltage differential between source and gate.
All in all, one half H-bridge home-grown to this recipe looks like this:
simulate this circuit – Schematic created using CircuitLab
Don't pay too much attention to the specific part numbers; there's quite a limited selection at CircuitLab.
Additionally, you'll need a logic-level N-channel device to switch the low end, as your logic control signal will only be 5V, and many power MOSFETs prefer a 10 V voltage to really turn on right. For driving 5A, that might not be so much of a problem if you get a high rated device.
You also want to try to time the switch-off of the low end to happen before the switch-on of the high end, else you may get cross conduction, which can be a quite nasty problem -- spikes of hundreds of amps, killing your H-bridge in no time! Been there, not pretty :-)
The diagram does not include this feature -- you'd have two separate control lines for the high end vs low end, rather than tying them together.
Finally, with 60V inductive load, you will see spikes up to 120V when trying to reverse polarity. This means you should have devices rated for 120V, which will be more expensive than your typical 30V 30A budget devices. It's worth the extra dollars, though, because a randomly failing H-bridge is not something you'll want to live with.
OK, so what you really want to do is to use a dedicated half-bridge driver circuit, hooked into the gates of MOSFET devices (IR2183, or similar,) as that device will do the cross-conduction prevention timing for you, and will properly drive the gates without unnecessarily burning current after switching like the pull-up resistor option will.
Or you want to buy a H-bridge already made and tested. Honestly, if this is just a one-off project, you'll be better of doing that, rather than building your own. Much less blood, sweat, and tears!
Or, finally, if your duty cycle is really low (switching only once a minute or whatever) then a DPDT relay will serve you just as well, and will be much easier to wire up!