I'm looking for a new 5k running app...

[cheapandcheerful]

...one that lets me play my own music and that records my time.

Any recommendations?

I think the nhs one does that.

If it's a C25K you're after, there's a free one from Zen Labs that rings each time you need to run, walk, warm up and cool down and you can play your music over it. If you're past C25K and just want to log 5K runs, MapMyWalk/Run etc. will alert you when you've done each kilometer as well as total time, average pace per kilometer and split pace per kilometer (you can also set it for miles and not to alert you to anything at all but rather just track it). You can also play your own music on it and it's free.

HTH