How many spades does my partner have when responding to my opening 1club bid and left overcalls 1 heart
Assuming that you have Negative Doubles, sometimes called Sputnik Doubles, in your arsenal of agreements, then partner is showing 5 or more Spades. With only four Spades partner would double instead.

The argument might be presented that partner is coming in with only 4 good spades as a lead directing call, but there is no reason for this. First, partner is odds on to have the opening lead if opponents win the auction; and second the double would also show a spade holding. Consequently this argument holds no water on this auction.

If you are not playing Negative Doubles then partner's call should either show five Spades and at least a minimum responding hand (normally 5+ HCP in standard systems), or four spades and a hand worth at least two free bids, normally 9+ HCP or so in standard systems). 

The reason for the two-way call is that with the stronger hand partner must make it clear that your side likely owns the auction, and can compete possibly to the 2- or 3-level.  You can determine after partner's next call if he might have only 4 Spades, with extra points. Initially assume partner is only showing 4 spades in this case, though it is acceptable to make a single raise to 2 with only three-card support if you are minimum.